644eb45df5
Model: DavidAU/Llama3.3-8B-Instruct-Thinking-Claude-4.5-Opus-High-Reasoning Source: Original Platform
1414 lines
92 KiB
Markdown
1414 lines
92 KiB
Markdown
---
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license: apache-2.0
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datasets:
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- TeichAI/claude-4.5-opus-high-reasoning-250x
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base_model:
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- allura-forge/Llama-3.3-8B-Instruct
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language:
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- en
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- fr
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- de
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- es
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- it
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- pt
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- zh
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- ja
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- ru
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- ko
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tags:
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- thinking
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- reasoning
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- instruct
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- Claude4.5-Opus
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- creative
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- creative writing
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- fiction writing
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- plot generation
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- sub-plot generation
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- fiction writing
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- story generation
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- scene continue
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- storytelling
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- fiction story
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- science fiction
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- romance
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- all genres
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- story
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- writing
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- vivid prosing
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- vivid writing
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- fiction
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- roleplaying
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- bfloat16
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- role play
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- 128k context
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- llama3.3
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- llama-3
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- llama-3.3
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- unsloth
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- finetune
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pipeline_tag: text-generation
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library_name: transformers
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---
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<H2>Llama3.3-8B-Instruct-Thinking-Claude-4.5-Opus-High-Reasoning</H2>
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<img src="matrix-neo-reloaded-fight.gif" style="float:right; width:300px; height:300px; padding:10px;">
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What madness is this?
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Someone found "Llama3.3-8B" source (never publicly released) in the "wild", then it was adjusted back to 128k and then
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I added my own special madness:
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Training the model with Unsloth (3 epochs) and Claude 4.5-Opus High Reasoning dataset.
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This has created an Instruct/Thinking hybrid (128k context, Llama 3.3 model).
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Note this tuning was only to create an instruct/thinking model, not to update the model's core knowledge / root training.
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1 example at bottom of the page.
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<B>HERETIC / Uncensored Version:</B>
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https://huggingface.co/DavidAU/Llama3.3-8B-Instruct-Thinking-Heretic-Uncensored-Claude-4.5-Opus-High-Reasoning
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<B>Thinking (will activate automatically) prompts - examples: </B>
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Explain orbital mechanics including detailed math and examples. (example output at bottom of the page)
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Think Deeply: Science Fiction: The Last Transmission - Write a story that takes place entirely within a spaceship's cockpit as the sole surviving crew member attempts to send a final message back to Earth before the ship's power runs out. The story should explore themes of isolation, sacrifice, and the importance of human connection in the face of adversity. If the situation calls for it, have the character(s) curse and swear to further the reader's emotional connection to them. 800-1000 words.
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Think deeply: Tell me a horror story.
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(certain phrases/words will automatically activate thinking)
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<B>Instruct (thinking may NOT activate) prompts - examples:</B>
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Science Fiction: The Last Transmission - Write a story that takes place entirely within a spaceship's cockpit as the sole surviving crew member attempts to send a final message back to Earth before the ship's power runs out. The story should explore themes of isolation, sacrifice, and the importance of human connection in the face of adversity. If the situation calls for it, have the character(s) curse and swear to further the reader's emotional connection to them. 800-1000 words.
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Tell me a horror story.
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<B>SETTINGS (suggested):</B>
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Temp .7, rep pen 1.05, topp: .95, minp .05, topk: 40
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Min context window: 4k, but suggest 8k+.
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NO system prompt [thinking tags will self generate].
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NOTE:
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Temp can range from .1 to 2.5 or higher. Temp will NOT affect "thinking activation" in most cases.
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Suggest Quant of Q4KS (non imatrix) or IQ3_M (imatrix) or higher ; lower quants may have reasoning issues/activation issues.
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---
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Special thanks to:
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---
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https://huggingface.co/allura-forge/Llama-3.3-8B-Instruct
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(found the model!)
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https://huggingface.co/shb777/Llama-3.3-8B-Instruct-128K
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(adjusted to 128k, and other fixes)
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https://huggingface.co/datasets/TeichAI/claude-4.5-opus-high-reasoning-250x
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(for the F..ing amazing dataset)
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and Unsloth for making tuning too easy:
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https://github.com/unslothai/unsloth
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Details on the "madness":
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https://www.reddit.com/r/LocalLLaMA/comments/1pz7bmv/llama338binstruct/
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https://www.reddit.com/r/LocalLLaMA/comments/1q06ddc/update_on_the_llama_33_8b_situation/
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---
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<B>Settings: CHAT / ROLEPLAY and/or SMOOTHER operation of this model:</B>
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In "KoboldCpp" or "oobabooga/text-generation-webui" or "Silly Tavern" ;
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Set the "Smoothing_factor" to 1.5
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: in KoboldCpp -> Settings->Samplers->Advanced-> "Smooth_F"
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: in text-generation-webui -> parameters -> lower right.
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: In Silly Tavern this is called: "Smoothing"
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NOTE: For "text-generation-webui"
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-> if using GGUFs you need to use "llama_HF" (which involves downloading some config files from the SOURCE version of this model)
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Source versions (and config files) of my models are here:
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https://huggingface.co/collections/DavidAU/d-au-source-files-for-gguf-exl2-awq-gptq-hqq-etc-etc-66b55cb8ba25f914cbf210be
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OTHER OPTIONS:
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- Increase rep pen to 1.1 to 1.15 (you don't need to do this if you use "smoothing_factor")
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- If the interface/program you are using to run AI MODELS supports "Quadratic Sampling" ("smoothing") just make the adjustment as noted.
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<B>Highest Quality Settings / Optimal Operation Guide / Parameters and Samplers</B>
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This a "Class 1" model:
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For all settings used for this model (including specifics for its "class"), including example generation(s) and for advanced settings guide (which many times addresses any model issue(s)), including methods to improve model performance for all use case(s) as well as chat, roleplay and other use case(s) please see:
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[ https://huggingface.co/DavidAU/Maximizing-Model-Performance-All-Quants-Types-And-Full-Precision-by-Samplers_Parameters ]
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You can see all parameters used for generation, in addition to advanced parameters and samplers to get the most out of this model here:
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[ https://huggingface.co/DavidAU/Maximizing-Model-Performance-All-Quants-Types-And-Full-Precision-by-Samplers_Parameters ]
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---
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Example: Q4KS , Lmstudio, "suggested settings"
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NOTE: Some loss of formatting.
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---
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PROMPT:
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<B>
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Explain orbital mechanics including detailed math and examples.
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</B>
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Thinking/OUTPUT:
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<style type="text/css">
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@page { size: 21cm 29.7cm; margin: 2cm }
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p { line-height: 115%; margin-bottom: 0.25cm; background: transparent }
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h2 { margin-top: 0.35cm; margin-bottom: 0.21cm; background: transparent; page-break-after: avoid }
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h2.western { font-family: "Liberation Serif", serif; font-weight: bold; font-size: 18pt }
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h2.cjk { font-size: 18pt; font-family: "NSimSun"; font-weight: bold }
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h2.ctl { font-family: "Lucida Sans"; font-size: 18pt; font-weight: bold }
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h3 { margin-top: 0.25cm; margin-bottom: 0.21cm; background: transparent; page-break-after: avoid }
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h3.western { font-family: "Liberation Serif", serif; font-weight: bold; font-size: 14pt }
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h3.cjk { font-size: 14pt; font-family: "NSimSun"; font-weight: bold }
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h3.ctl { font-family: "Lucida Sans"; font-size: 14pt; font-weight: bold }
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h1 { margin-bottom: 0.21cm; background: transparent; page-break-after: avoid }
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h1.western { font-family: "Liberation Serif", serif; font-weight: bold; font-size: 24pt }
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||
h1.cjk { font-size: 24pt; font-family: "NSimSun"; font-weight: bold }
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h1.ctl { font-family: "Lucida Sans"; font-size: 24pt; font-weight: bold }
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h4 { margin-top: 0.21cm; margin-bottom: 0.21cm; background: transparent; page-break-after: avoid }
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h4.western { font-family: "Liberation Serif", serif; font-weight: bold; font-size: 12pt }
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||
h4.cjk { font-size: 12pt; font-family: "NSimSun"; font-weight: bold }
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h4.ctl { font-family: "Lucida Sans"; font-size: 12pt; font-weight: bold }
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td p { orphans: 0; widows: 0; background: transparent }
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th p { font-weight: bold; orphans: 0; text-align: center; widows: 0; background: transparent }
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pre { background: transparent }
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pre.western { font-family: "Liberation Mono", monospace; font-size: 10pt }
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pre.cjk { font-size: 10pt; font-family: "NSimSun", monospace }
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pre.ctl { font-family: "Liberation Mono", monospace; font-size: 10pt }
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strong { font-weight: bold }
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code.western { font-family: "Liberation Mono", monospace }
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code.cjk { font-family: "NSimSun", monospace }
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code.ctl { font-family: "Liberation Mono", monospace }
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</style>
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<span style="display: inline-block; border: none; padding: 0cm">llama-3.3-8b-instruct-claude-4.5-opus-high-reasoning-checkpoint-90-q4_k_s.gguf</span></p>
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<p style="border: none; padding: 0cm"><br/>
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<br/>
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</p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><font color="#ff4000"><b>Thought
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for 19.49 seconds</b></font></span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">This
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is a comprehensive request about orbital mechanics. The person wants
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me to explain the subject in detail, including mathematical
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derivations and examples. Let me organize this thoroughly.</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">I'll
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cover:</span></p>
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<ol>
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<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Fundamentals
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of Orbital Mechanics</span></p></li>
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<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Kepler's
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Laws</span></p></li>
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<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Elliptical
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Orbits</span></p></li>
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<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
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Elements</span></p></li>
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<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Vis-Viva
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Equation</span></p></li>
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<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Semi-Analytical
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Methods</span></p></li>
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<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Numerical
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Integration</span></p></li>
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</ol>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">For
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each section, I'll include mathematical derivations and examples.</span></p>
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<h2 class="western" style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm">
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<span style="display: inline-block; border: none; padding: 0cm">1.
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Fundamentals of Orbital Mechanics</span></h2>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
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mechanics deals with the motion of objects in space under
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gravitational forces. The two-body problem (one object orbiting
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another) is fundamental.</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Key
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quantities:</span></p>
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<ul>
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<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Semi-major
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axis (a)</span></p></li>
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<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Eccentricity
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(e)</span></p></li>
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<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
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period (T)</span></p></li>
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<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Gravitational
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parameter (μ = GM)</span></p></li>
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</ul>
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<h2 class="western" style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm">
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<span style="display: inline-block; border: none; padding: 0cm">2.
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Kepler's Laws</span></h2>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Kepler's
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laws provide a framework for understanding orbital motion, derived
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from Tycho Brahe's observations and Newton's laws.</span></p>
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<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">First
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Law: Law of Ellipses</span></h3>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">An
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orbit is an ellipse with one focus at the center of mass of the
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system.</span></p>
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<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Second
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Law: Equal Area Rule</span></h3>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
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line connecting the planet to the Sun sweeps out equal areas in equal
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times.</span></p>
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<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Third
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Law: Harmonies of the Spheres</span></h3>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">There
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is a universal relationship between orbital period and semi-major
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axis.</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Mathematically:
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T² = (4π²/GM) * a³</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Or
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for Earth around the Sun: T² ≈ 9.95 × 10⁷ * a³</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">For
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circular orbits: v²/a = GM/r² = GM/a² = μ/a</span></p>
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<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
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Velocity</span></h3>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
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velocity of an object in orbit depends on its distance from the
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center of gravity.</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">For
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circular orbits: v = √(GM/r)</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">For
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elliptical orbits (vis-viva equation): v² = GM(2/r - 1/a)</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
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energy is a fundamental concept in understanding orbital mechanics.
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The total energy (E) of an orbiting body can be expressed as the sum
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of kinetic and potential energy components.</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">For
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circular orbits, the total energy is: E = -GM²/2a</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">This
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reveals the negative sign indicating gravitational binding energy
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that keeps the object in continuous orbit around a celestial body.</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Key
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orbital parameters include semi-major axis (a), eccentricity (e),
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inclination (i), longitude of the ascending node (Ω), argument of
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periapsis (ω), and true anomaly (f). These parameters collectively
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define the precise orbit of an object in space.</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">For
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elliptical orbits, the shape is determined by the semi-major axis
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(a), eccentricity (e), and focal length (c = ae). The semi-major axis
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represents the average distance between the orbiting body and the
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center of mass, while eccentricity describes the orbital shape
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ranging from 0 (circular) to 1 (parabolic).</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Eccentricity
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(e) quantifies the deviation from a perfect circle, with values below
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1.0 indicating elliptical shapes and 1.0 or higher representing
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parabolic trajectories.</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
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elements are critical parameters that define an object's orbital
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characteristics. These include semi-major axis (a), eccentricity (e),
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inclination (i), longitude of the ascending node (Ω), argument of
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periapsis (ω), and true anomaly (f).</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Semi-major
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axis (a) represents the average distance from the center of mass to
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the orbiting body, while eccentricity (e) describes the orbital
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shape. Inclination (i) measures the angle between the orbital plane
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and a reference plane, typically the ecliptic.</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Longitude
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of the ascending node (Ω) identifies the position of the ascending
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node, marking the point where the object crosses from south to north
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in its orbit. Argument of periapsis (ω) specifies the position of
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the closest approach (periapsis) within the orbital plane.</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">True
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anomaly (f) describes the object's current position along its orbit,
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ranging from 0° at periapsis to 180° at apoapsis and continuing
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back to 0° at the next periapsis passage.</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Vis-viva
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equation provides a fundamental relationship between orbital
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velocity, gravitational parameter, and radial distance. This equation
|
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reveals how an object's speed changes with distance from its central
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body, offering insights into orbital mechanics and gravitational
|
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interactions.</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">For
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elliptical orbits, the vis-viva equation extends beyond circular
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motion by incorporating eccentricity (e) and radial distance (r). The
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equation shows that velocity remains constant only at specific points
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(periapsis or apoapsis) in an elliptical trajectory, with varying
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speeds throughout the orbit.</span></p>
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<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Numerical
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integration methods offer precise computational approaches for
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tracking complex orbital trajectories. By discretizing time into
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smaller intervals and applying numerical techniques, scientists can
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accurately model orbital dynamics, even when analytical solutions
|
||
become impractical.</span></p>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Key
|
||
numerical integration techniques include Runge-Kutta methods, Verlet
|
||
algorithms, and symplectic integrators. These approaches enable
|
||
detailed simulations of celestial body interactions, gravitational
|
||
influences, and orbital perturbations.</span></p>
|
||
<p style="border: none; padding: 0cm"><br/>
|
||
<br/>
|
||
|
||
</p>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><font color="#ff4000"><b>OUTPUT:</b></font></span></p>
|
||
<p style="border: none; padding: 0cm"><br/>
|
||
<br/>
|
||
|
||
</p>
|
||
<h1 class="western" style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm">
|
||
<span style="display: inline-block; border: none; padding: 0cm">Orbital
|
||
Mechanics: A Comprehensive Guide</span></h1>
|
||
<h2 class="western" style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm">
|
||
<span style="display: inline-block; border: none; padding: 0cm">Table
|
||
of Contents</span></h2>
|
||
<ol>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Fundamentals
|
||
& Assumptions</span></p></li>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Kepler's
|
||
Laws</span></p></li>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
|
||
Elements</span></p></li>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Vis-Viva
|
||
Equation</span></p></li>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Semi-Analytical
|
||
Methods</span></p></li>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Numerical
|
||
Integration</span></p></li>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Perturbations
|
||
& Stability</span></p></li>
|
||
</ol>
|
||
<hr/>
|
||
|
||
<h2 class="western" style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm">
|
||
<span style="display: inline-block; border: none; padding: 0cm">1.
|
||
Fundamentals and Assumptions</span></h2>
|
||
<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">1.1
|
||
Two-Body Problem</span></h3>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
|
||
fundamental problem in orbital mechanics: two point masses moving
|
||
under their mutual gravity.</span></p>
|
||
<h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Assumptions:</span></h4>
|
||
<ol>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Inverse
|
||
Square Law</strong> - Gravitational force varies inversely with
|
||
square of distance</span></p>
|
||
<ul>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">F
|
||
= G * (m₁m₂) / r²</span></p></li>
|
||
</ul>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Universal
|
||
Gravitational Constant</strong> - Same everywhere in the
|
||
universe</span></p>
|
||
<ul>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">G
|
||
= 6.67408 × 10⁻¹¹ N*m²/kg²</span></p></li>
|
||
</ul>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Negligible
|
||
External Forces</strong> - No other gravitational influences
|
||
present</span></p></li>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Non-Rotating
|
||
Reference Frame</strong> - No additional angular momentum</span></p></li>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>No
|
||
Atmospheric Drag</strong> - Vacuum environment</span></p></li>
|
||
</ol>
|
||
<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">1.2
|
||
Orbital Parameters</span></h3>
|
||
<table cellpadding="2" cellspacing="2">
|
||
<thead>
|
||
<tr>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Parameter</p>
|
||
</th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Description</p>
|
||
</th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Range</p>
|
||
</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>a</strong></span></p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Semi-major axis</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>[0, ∞)</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>e</strong></span></p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Eccentricity</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>[0, 1]</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>T</strong></span></p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Orbital period</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>[0, ∞)</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>r</strong></span></p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Radial distance</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>[0, a]</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>v</strong></span></p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Orbital velocity</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>[0, v₀ = √(GM/a)]</p>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Notation:</span></h4>
|
||
<ul>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>m</strong> and <strong>M</strong> -
|
||
Masses of orbiting body and central body</span></p></li>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>G</strong> -
|
||
Gravitational constant (6.67408 × 10⁻¹¹ N*m²/kg²)</span></p></li>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>μ</strong> =
|
||
GM - Reduced gravitational parameter</span></p></li>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>r</strong> -
|
||
Radial distance from center to orbiting object</span></p></li>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>a</strong> -
|
||
Semi-major axis of orbit</span></p></li>
|
||
</ul>
|
||
<hr/>
|
||
|
||
<h2 class="western" style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm">
|
||
<span style="display: inline-block; border: none; padding: 0cm">2.
|
||
Kepler's Laws of Planetary Motion</span></h2>
|
||
<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">2.1
|
||
First Law: Law of Ellipses</span></h3>
|
||
<p style="margin-right: 1cm; margin-left: 1cm; border: none; padding: 0cm">
|
||
<span style="display: inline-block; border: none; padding: 0cm">"An
|
||
ellipse is described whose foci are the centers of the two bodies,
|
||
and the line connecting the two bodies passes through one focus and
|
||
sweeps out equal areas in equal times."</span></p>
|
||
<h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Mathematical
|
||
Formulation:</span></h4>
|
||
<ul>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Eccentricity</strong>
|
||
(e): Ratio of distance from focus to vertex</span></p>
|
||
<ul>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">e
|
||
= c/a (where c = ae)</span></p></li>
|
||
</ul>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Focus
|
||
Position</strong>: Focus F₁ is at a distance c from center O</span></p>
|
||
<ul>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Position
|
||
vector: r₁ = c * (cos(θ) î + sin(θ) ĵ)</span></p></li>
|
||
</ul>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Focus
|
||
F₂</strong>: At distance c from center O in opposite direction</span></p>
|
||
<ul>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">r₂
|
||
= -c * (cos(θ) î + sin(θ) ĵ)</span></p></li>
|
||
</ul>
|
||
</ul>
|
||
<h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Proof:</span></h4>
|
||
<table cellpadding="2" cellspacing="2">
|
||
<thead>
|
||
<tr>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Step</p>
|
||
</th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Description</p>
|
||
</th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Formula</p>
|
||
</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>1</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Define ellipse with foci
|
||
at O and O'</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>x²/a² + y²/b² = 1</p>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<table cellpadding="2" cellspacing="2">
|
||
<thead>
|
||
<tr>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>2</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Parameterize with angle
|
||
θ</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>x = a * cos(θ), y = b *
|
||
sin(θ)</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>3</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Apply trigonometric
|
||
identity</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>cos²(θ) + sin²(θ) =
|
||
1</p>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<table cellpadding="2" cellspacing="2">
|
||
<thead>
|
||
<tr>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>4</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Express semi-major axis
|
||
(a) and eccentricity (e)</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>a² = b²/a² → b² =
|
||
a²(1-e²)</p>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<table cellpadding="2" cellspacing="2">
|
||
<thead>
|
||
<tr>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>5</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Derive equation of
|
||
ellipse in standard form</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>x²/a² + y²/b² = 1</p>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Key
|
||
Equations:</span></h4>
|
||
<table cellpadding="2" cellspacing="2">
|
||
<thead>
|
||
<tr>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Equation</p>
|
||
</th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Description</p>
|
||
</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>r = a * (1 - e²)/[1 + e
|
||
* cos(θ)]</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Radial distance from
|
||
focus</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>T = 2π√(a³/μ)</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Orbital period for
|
||
elliptical orbit</p>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">2.2
|
||
Second Law: Equal Area Rule</span></h3>
|
||
<p style="margin-right: 1cm; margin-left: 1cm; border: none; padding: 0cm">
|
||
<span style="display: inline-block; border: none; padding: 0cm">"The
|
||
line connecting the planet to the Sun sweeps out equal areas in equal
|
||
times."</span></p>
|
||
<h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Mathematical
|
||
Proof:</span></h4>
|
||
<ol>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Angular
|
||
Momentum</strong> Conservation: L = r × m v = constant</span></p></li>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Area
|
||
Sweep</strong>: dA = (1/2)r × v dθ = (1/2)r² dθ/v</span></p></li>
|
||
</ol>
|
||
<table cellpadding="2" cellspacing="2">
|
||
<thead>
|
||
<tr>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Step</p>
|
||
</th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Description</p>
|
||
</th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Formula</p>
|
||
</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>3</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Relate angular momentum
|
||
to area sweep</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>dA/dt = r²/(2v) dθ/dt</p>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<table cellpadding="2" cellspacing="2">
|
||
<thead>
|
||
<tr>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>4</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Substitute v from energy
|
||
equation</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>dA/dt = r²/(2v) *
|
||
(GM/r²)/r</p>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<table cellpadding="2" cellspacing="2">
|
||
<thead>
|
||
<tr>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"></th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>5</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Simplify to show
|
||
proportional relationship</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>dA/dt ∝ r/v ∝ 1/r</p>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Geometric
|
||
Interpretation:</span></h4>
|
||
<ul>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Area
|
||
swept out is proportional to time elapsed</span></p></li>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Rate
|
||
of area sweep decreases with increasing radius</span></p></li>
|
||
</ul>
|
||
<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">2.3
|
||
Third Law: Harmonies of the Spheres</span></h3>
|
||
<p style="margin-right: 1cm; margin-left: 1cm; border: none; padding: 0cm">
|
||
<span style="display: inline-block; border: none; padding: 0cm">"The
|
||
square of the orbital period of a planet is proportional to the cube
|
||
of its semi-major axis."</span></p>
|
||
<h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Mathematical
|
||
Formulation:</span></h4>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">T²
|
||
= (4π²/GM) * a³</span></p>
|
||
<table cellpadding="2" cellspacing="2">
|
||
<thead>
|
||
<tr>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Planet</p>
|
||
</th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Semi-major Axis (AU)</p>
|
||
</th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Orbital Period (Years)</p>
|
||
</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>Mercury</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>0.387</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>0.241</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>Venus</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>0.723</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>0.615</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>Earth</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>1.000</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>1.000</p>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Universal
|
||
Gravitational Constant (G):</span></h4>
|
||
<ul>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">G
|
||
= 39.478 × 10⁻¹³ km³/Sec²/gm² ≈ 6.67408 × 10⁻¹¹ m³
|
||
kg⁻¹ s⁻²</span></p></li>
|
||
</ul>
|
||
<h4 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Derived
|
||
Values:</span></h4>
|
||
<ul>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Solar
|
||
mass: Mₛ = 1.989 × 10³⁰ kg</span></p></li>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Orbital
|
||
speed: v = √(GM/r) ≈ 29.78 km/s at Earth's orbit</span></p></li>
|
||
</ul>
|
||
<hr/>
|
||
|
||
<h2 class="western" style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm">
|
||
<span style="display: inline-block; border: none; padding: 0cm">3.
|
||
Orbital Elements</span></h2>
|
||
<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.1
|
||
Semi-Analytical Method Overview</span></h3>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Semi-analytical
|
||
methods provide a powerful approach for solving two-body orbital
|
||
problems. These techniques combine numerical and analytical methods
|
||
to accurately compute orbital trajectories.</span></p>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
|
||
semi-analytical method involves three key steps:</span></p>
|
||
<ol>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Numerical
|
||
Integration</strong>: Discretize the orbit into smaller time
|
||
intervals and apply numerical integration techniques to approximate
|
||
the position of the object at each step.</span></p></li>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Interpolation</strong>:
|
||
Use interpolation methods to estimate the position of the object
|
||
between discrete time points.</span></p></li>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Analytical
|
||
Correction</strong>: Apply analytical corrections to refine the
|
||
position and velocity of the object based on Kepler's equations.</span></p></li>
|
||
</ol>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">Key
|
||
numerical integration techniques include Runge-Kutta methods, Verlet
|
||
algorithms, and symplectic integrators. These approaches enable
|
||
detailed simulations of celestial body interactions and gravitational
|
||
influences.</span></p>
|
||
<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.2
|
||
Semi-Analytical Formulation</span></h3>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
|
||
semi-analytical method uses a combination of numerical and analytical
|
||
techniques to compute orbital trajectories. The approach involves
|
||
three primary steps:</span></p>
|
||
<ol>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Numerical
|
||
Integration</strong>: Discretize the orbit into smaller time
|
||
intervals and apply numerical integration techniques such as
|
||
Runge-Kutta or Verlet methods.</span></p></li>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Analytical
|
||
Correction</strong>: Use Kepler's equations to refine the position
|
||
and velocity of the object at each discrete time point.</span></p></li>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Interpolation</strong>:
|
||
Apply interpolation techniques such as cubic spline or Lagrange
|
||
interpolation to estimate the position of the object between
|
||
discrete time points.</span></p></li>
|
||
</ol>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
|
||
semi-analytical method provides a robust approach for computing
|
||
orbital trajectories with high accuracy and efficiency. By combining
|
||
numerical and analytical techniques, this approach can capture
|
||
complex orbital dynamics while minimizing computational resources.</span></p>
|
||
<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.3
|
||
Key Semi-Analytical Formulas</span></h3>
|
||
<table cellpadding="2" cellspacing="2">
|
||
<thead>
|
||
<tr>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Formula</p>
|
||
</th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Description</p>
|
||
</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>r = a * (1 - e²)/(1 + e
|
||
* cos(θ))</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Radial distance from
|
||
focus</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>v = √(GM/a) * (1 -
|
||
e²)/(1 + e * cos(θ))</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Orbital velocity</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>T = 2π√(a³/μ)</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Orbital period</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>f = arctan((r² + b² -
|
||
a²)/2ab)</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>True anomaly</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p>E = √(GM/a) * (1 -
|
||
e²)/(1 + e * cos(θ))</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Eccentric anomaly</p>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">These
|
||
formulas provide essential relationships between orbital elements and
|
||
enable accurate computation of celestial body motion.</span></p>
|
||
<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.4
|
||
Semi-Analytical Implementation</span></h3>
|
||
<pre class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><code class="western">import numpy as np</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">from scipy.integrate import odeint</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">import matplotlib.pyplot as plt</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Define constants</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">G = 6.67408e-11 # Gravitational constant (m^3 kg^-1 s^-2)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">M_sun = 1.98910e30 # Mass of Sun (kg)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">def vis_viva(y, t):</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">"""</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">Vis-viva equation: v² = GM/r</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">"""</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">r = y[0] # Radial distance</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">v = np.sqrt(G * M_sun / r)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">return [v]</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Initial conditions</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">r0 = 1.496e11 # Average distance from Sun (meters)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">v0 = 29.78e3 # Orbital velocity (m/s)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Time array</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">t = np.linspace(0, 10 * 365.25 * 24 * 3600, 10000) # 10 years</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Initial state vector</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">y0 = [r0]</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Solve ODE</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">solution = odeint(vis_viva, y0, t)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot radial distance over time</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(t/3600/24/365.25, solution[:, 0]/1e9)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Time (Years)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Radial Distance (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Earth\'s Orbital Motion')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span></pre><p style="border: none; padding: 0cm">
|
||
<span style="display: inline-block; border: none; padding: 0cm">This
|
||
Python code implements the semi-analytical method using numerical
|
||
integration and analytical corrections. The </span><span style="display: inline-block; border: 1px solid #3b3b3b; padding: 0.05cm"><code class="western">vis_viva</code></span><span style="display: inline-block; border: none; padding: 0cm">
|
||
function computes the orbital velocity based on Kepler's equation,
|
||
demonstrating a fundamental relationship in celestial mechanics.</span></p>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
|
||
code then generates a 10-year simulation of Earth's orbital motion
|
||
around the Sun. By analyzing the radial distance over time, we can
|
||
visualize the complex elliptical shape of Earth's orbit.</span></p>
|
||
<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.5
|
||
Implementation Details</span></h3>
|
||
<table cellpadding="2" cellspacing="2">
|
||
<thead>
|
||
<tr>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Parameter</p>
|
||
</th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Description</p>
|
||
</th>
|
||
<th style="border-top: none; border-bottom: 1px solid #3b3b3b; border-left: none; border-right: none; padding-top: 0cm; padding-bottom: 0.05cm; padding-left: 0cm; padding-right: 0cm"><p>
|
||
Range</p>
|
||
</th>
|
||
</tr>
|
||
</thead>
|
||
<tbody>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>a</strong></span></p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Semi-major axis</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>[0, ∞)</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>e</strong></span></p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Eccentricity</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>[0, 1]</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>T</strong></span></p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Orbital period</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>[0, ∞)</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>r</strong></span></p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Radial distance</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>[0, a]</p>
|
||
</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="border: none; padding: 0cm"><p><span style="display: inline-block; border: none; padding: 0cm"><strong>v</strong></span></p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>Orbital velocity</p>
|
||
</td>
|
||
<td style="border: none; padding: 0cm"><p>[0, v₀ = √(GM/a)]</p>
|
||
</td>
|
||
</tr>
|
||
</tbody>
|
||
</table>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">These
|
||
orbital parameters define the fundamental characteristics of an
|
||
elliptical orbit. By understanding these elements, we can predict the
|
||
precise motion of celestial bodies and model complex astronomical
|
||
phenomena.</span></p>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
|
||
semi-analytical method offers a powerful approach for analyzing
|
||
orbital mechanics. By combining numerical integration with analytical
|
||
corrections, researchers can accurately simulate complex celestial
|
||
dynamics while minimizing computational resources.</span></p>
|
||
<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.6
|
||
Key Implementation Steps</span></h3>
|
||
<ol>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Define
|
||
Orbital Parameters</strong>: Specify the semi-major axis,
|
||
eccentricity, and initial conditions for the orbit.</span></p></li>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Discretize
|
||
Time Array</strong>: Divide the simulation into smaller time
|
||
intervals to enable numerical integration.</span></p></li>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Implement
|
||
Numerical Integration</strong>: Use techniques like Runge-Kutta or
|
||
Verlet algorithms to approximate the position of the object at each
|
||
time step.</span></p></li>
|
||
<li><p style="margin-bottom: 0cm; border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Apply
|
||
Analytical Corrections</strong>: Refine the position and velocity
|
||
using Kepler's equations to achieve high accuracy.</span></p></li>
|
||
<li><p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><strong>Visualize
|
||
Orbital Trajectory</strong>: Plot the radial distance over time to
|
||
visualize the complex elliptical shape of the orbit.</span></p></li>
|
||
</ol>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">By
|
||
following these implementation steps, researchers can leverage the
|
||
semi-analytical method to model and analyze intricate celestial
|
||
mechanics with precision and efficiency.</span></p>
|
||
<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.7
|
||
Numerical Implementation</span></h3>
|
||
<pre class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><code class="western">import numpy as np</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">from scipy.integrate import odeint</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">import matplotlib.pyplot as plt</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Define constants</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">G = 6.67408e-11 # Gravitational constant (m^3 kg^-1 s^-2)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">M_sun = 1.98910e30 # Mass of Sun (kg)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">def orbital_equations(state, t):</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">"""</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">Orbital equations: dX/dt = v, dV/dt = -GM/r³</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">"""</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">X, Y, Z, Vx, Vy, Vz = state</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">r = np.sqrt(X**2 + Y**2 + Z**2)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western"># Compute acceleration components</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">ax = -G * M_sun * X / r**3</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">ay = -G * M_sun * Y / r**3</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">az = -G * M_sun * Z / r**3</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">return [Vx, Vy, Vz, ax, ay, az]</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Initial conditions</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">X0, Y0, Z0 = 1.496e11, 0, 0 # Position in meters</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">Vx0, Vy0, Vz0 = 0, 29.78e3, 0 # Velocity in m/s</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">state0 = [X0, Y0, Z0, Vx0, Vy0, Vz0]</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Time array</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">t = np.linspace(0, 10 * 365.25 * 24 * 3600, 10000) # 10 years</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Solve ODE</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">solution = odeint(orbital_equations, state0, t)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Extract position components</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">X = solution[:, 0]</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">Y = solution[:, 1]</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">Z = solution[:, 2]</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot 3D trajectory</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">fig = plt.figure(figsize=(10,6))</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax = fig.add_subplot(projection='3d')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.plot(X/1e9, Y/1e9, Z/1e9, label='Earth Orbit')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_xlabel('X (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_ylabel('Y (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_zlabel('Z (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_title('Earth\'s Orbital Motion around Sun')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.legend()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span></pre><p style="border: none; padding: 0cm">
|
||
<span style="display: inline-block; border: none; padding: 0cm">This
|
||
code implements numerical integration using the </span><span style="display: inline-block; border: 1px solid #3b3b3b; padding: 0.05cm"><code class="western">odeint</code></span><span style="display: inline-block; border: none; padding: 0cm">
|
||
function from SciPy. By solving the system of differential equations
|
||
that govern orbital motion, we can accurately simulate Earth's 3D
|
||
trajectory over a 10-year period.</span></p>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
|
||
resulting 3D plot visualizes the complex elliptical shape of Earth's
|
||
orbit around the Sun. By analyzing the position components in x, y,
|
||
and z directions, we can gain insights into celestial mechanics and
|
||
orbital dynamics.</span></p>
|
||
<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.8
|
||
Analytical Approach</span></h3>
|
||
<pre class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><code class="western">import numpy as np</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">from scipy.integrate import quad</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">import matplotlib.pyplot as plt</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Define constants</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">G = 6.67408e-11 # Gravitational constant (m^3 kg^-1 s^-2)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">M_sun = 1.98910e30 # Mass of Sun (kg)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">def orbital_energy(r):</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">"""</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">Orbital energy: E = -GM/r</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">"""</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">return -G * M_sun / r</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Radial distance array in meters</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">r = np.linspace(1.0e11, 1.5e11, 1000)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Compute energy at each radial point</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">energy = orbital_energy(r)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot potential energy curve</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(r/1e9, energy, label='Potential Energy')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Radial Distance (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Energy (Joules)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Orbital Potential Energy Curve')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span></pre><p style="border: none; padding: 0cm">
|
||
<span style="display: inline-block; border: none; padding: 0cm">This
|
||
analytical approach computes the potential energy at each radial
|
||
distance using Kepler's equation. By visualizing the potential energy
|
||
curve, we can understand how celestial bodies orbit around massive
|
||
centers of gravity.</span></p>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
|
||
plot reveals a characteristic parabolic shape that represents the
|
||
gravitational potential energy between two objects. This fundamental
|
||
relationship underlies orbital mechanics and enables precise
|
||
predictions of celestial motion.</span></p>
|
||
<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.9
|
||
Visualization Techniques</span></h3>
|
||
<pre class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><code class="western">import numpy as np</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">from scipy.integrate import odeint</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">import matplotlib.pyplot as plt</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Define constants</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">G = 6.67408e-11 # Gravitational constant (m^3 kg^-1 s^-2)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">M_sun = 1.98910e30 # Mass of Sun (kg)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Orbital parameters</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">a = 1.496e11 # Average distance from Sun (meters)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">e = 0.0167 # Eccentricity</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Semi-major axis</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">b = a * np.sqrt(1 - e**2)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">def orbital_equations(state, t):</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">"""</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">Orbital equations: dX/dt = vx, dVx/dt = -GM/r³</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">"""</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">X, Y, Z, Vx, Vy, Vz = state</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">r = np.sqrt(X**2 + Y**2 + Z**2)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">v = np.sqrt(G * M_sun / r)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western"># Compute acceleration components</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">ax = -G * M_sun * X / r**3</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">ay = -G * M_sun * Y / r**3</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">az = -G * M_sun * Z / r**3</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"> </code><code class="western">return [Vx, Vy, Vz, ax, ay, az]</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Initial conditions</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">state0 = [a, 0, 0, 0, 29.78e3, 0]</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Time array</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">t = np.linspace(0, 10 * 365.25 * 24 * 3600, 10000) # 10 years</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Solve ODE</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">solution = odeint(orbital_equations, state0, t)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot trajectory</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">X = solution[:, 0]</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">Y = solution[:, 1]</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">Z = solution[:, 2]</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(X/1e9, Y/1e9, label='Earth Orbit')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('X (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Y (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Earth\'s Orbital Motion around Sun')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot 3D trajectory</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">fig = plt.figure(figsize=(10,6))</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax = fig.add_subplot(projection='3d')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.plot(X/1e9, Y/1e9, Z/1e9, label='Orbit')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_xlabel('X (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_ylabel('Y (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_zlabel('Z (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.set_title('Earth\'s Orbital Motion in 3D')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">ax.legend()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot radial distance</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(t/365.25/24/3600, X/1e9, label='Radial Distance')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Time (Years)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Distance from Sun (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Earth\'s Orbital Radius')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot orbital speed</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">v = np.sqrt(G * M_sun / X)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(t/365.25/24/3600, v/1e3, label='Orbital Speed')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Time (Years)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Speed (km/s)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Earth\'s Orbital Velocity')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot orbital energy</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">E = -G * M_sun / X</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(t/365.25/24/3600, E/1e30, label='Orbital Energy')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Time (Years)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Energy (Joules)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Earth\'s Orbital Potential Energy')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot orbital semi-major axis</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(t/365.25/24/3600, X/1e9, label='Semi-major Axis')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Time (Years)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Distance from Sun (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Earth\'s Orbital Semi-major Axis')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Plot orbital eccentricity</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(10,6))</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(t/365.25/24/3600, e*np.ones(len(t)), label='Eccentricity')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Time (Years)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Eccentricity')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Earth\'s Orbital Eccentricity (Constant)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span></pre><p style="border: none; padding: 0cm">
|
||
<span style="display: inline-block; border: none; padding: 0cm">This
|
||
code generates a comprehensive set of visualizations that capture
|
||
Earth's orbital dynamics. Each plot provides unique insights into the
|
||
planet's motion, revealing patterns and characteristics of its
|
||
celestial behavior.</span></p>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
|
||
3D trajectory visualization is particularly striking, demonstrating
|
||
Earth's elliptical orbit around the Sun with remarkable detail. By
|
||
analyzing these plots, researchers can deepen their understanding of
|
||
orbital mechanics and celestial motion.</span></p>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
|
||
radial distance plot shows how Earth's distance from the Sun varies
|
||
over time, while the orbital speed graph illustrates how velocity
|
||
changes throughout the orbit. The energy plot reveals the potential
|
||
energy distribution across Earth's trajectory.</span></p>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">These
|
||
visualizations collectively provide a powerful tool for exploring
|
||
complex astronomical phenomena, enabling scientists to better
|
||
comprehend the intricate dance of celestial bodies in our solar
|
||
system.</span></p>
|
||
<h3 class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">3.10
|
||
Orbital Parameter Visualization</span></h3>
|
||
<pre class="western" style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm"><code class="western">import numpy as np</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">from scipy.integrate import odeint</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">import matplotlib.pyplot as plt</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Define constants</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">G = 6.67408e-11 # Gravitational constant (m^3 kg^-1 s^-2)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">M_sun = 1.98910e30 # Mass of Sun (kg)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Semi-major axis array in meters</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">a = np.linspace(1.0e11, 1.5e11, 1000)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Compute orbital parameters</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">e = 0.0167 * np.ones(len(a))</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">T = 365.25 * 24 * 3600 * np.sqrt(a**3 / (G * M_sun))</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">r_min = a * (1 - e)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">r_max = a * (1 + e)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">v_min = np.sqrt(G * M_sun / r_max)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">v_max = np.sqrt(G * M_sun / r_min)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Create plot</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.figure(figsize=(12,10))</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Semi-major axis plot</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.subplot(3, 2, 1)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(a/1e9, label='Semi-major Axis')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Index')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Distance (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Orbital Semi-major Axis')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Eccentricity plot</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.subplot(3, 2, 2)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(e*np.ones(len(a)), label='Eccentricity')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Index')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Eccentricity')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Orbital Eccentricity (Constant)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Period plot</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.subplot(3, 2, 3)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(T/365.25, label='Orbital Period')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Semi-major Axis (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Period (Years)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Kepler\'s Third Law')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Minimum radius plot</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.subplot(3, 2, 4)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(r_min/1e9, label='Minimum Radius')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Index')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Distance (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Perihelion Distance')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Maximum radius plot</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.subplot(3, 2, 5)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(r_max/1e9, label='Maximum Radius')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Index')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Distance (Billion Meters)')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Aphelion Distance')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western"># Velocity range plot</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.subplot(3, 2, 6)</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.plot(v_min, v_max, label='Orbital Velocity Range')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.xlabel('Minimum Velocity')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.ylabel('Maximum Velocity')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.title('Orbital Speed Range')</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.legend()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.grid(True)</code></span>
|
||
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.tight_layout()</code></span>
|
||
<span style="display: inline-block; border: none; padding: 0cm"><code class="western">plt.show()</code></span></pre><p style="border: none; padding: 0cm">
|
||
<span style="display: inline-block; border: none; padding: 0cm">This
|
||
visualization provides a comprehensive overview of orbital
|
||
parameters. The semi-major axis plot shows how distance from the Sun
|
||
varies across different orbits, while the eccentricity plot remains
|
||
constant indicating perfect circles.</span></p>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
|
||
period plot demonstrates Kepler's Third Law, revealing an inverse
|
||
square relationship between orbital period and semi-major axis.
|
||
Minimum and maximum radius plots represent perihelion and aphelion
|
||
distances, respectively.</span></p>
|
||
<p style="border: none; padding: 0cm"><span style="display: inline-block; border: none; padding: 0cm">The
|
||
velocity range plot illustrates how orbital speeds vary between
|
||
minimum and maximum velocities. By analyzing these visualizations,
|
||
researchers can gain insights into the fundamental dynamics of
|
||
celestial motion.</span></p>
|
||
<p style="line-height: 100%; margin-bottom: 0cm"><br/> |