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本工具的運算數值是否具備學術與專業參考價值？

是的。所有運算演算法均基於所屬學科公認之國際標準公式編寫，經過嚴格的邊界極限測試，能夠滿足日常工作、課堂學習與工程初步核算之精度要求。

遇到特殊邊界數值或計算結果異常，該如何處理？

請檢查輸入的各項參數單位是否統一、數值是否落於常規物理或邏輯有效區間內（例如數值不能為負數、分母不能為零）。您亦可點選「重設參數」重置為基準值再次測算。

我可以將測算出的結果保存或分享給團隊嗎？

可以。本工具配備「📋 複製結果」與「🖨️ 列印報表」按鈕，點選即可將數據複製到剪貼簿，亦可直接列印或另存為高清晰度 PDF 文件。

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orbital 速度與速率計算機｜線上即時試算、公式解析與操作指南

What Is Orbital Velocity?

Orbital velocity is the speed an object must travel to stay in a stable circular orbit around a more massive body, such as a satellite around Earth or a planet around the Sun. At this speed, the gravitational pull of the central body exactly supplies the centripetal force needed to keep the object curving around it. Too slow and it falls inward; too fast and it escapes.

The Formula

For a circular orbit, orbital velocity is given by

v = \sqrt{\dfrac{G · M}{r}}

where G is the gravitational constant (6.674 × 10^{-11}\ N·m^2/kg^2), M is the mass of the central body in kilograms, and r is the orbital radius in meters (measured from the center of the central body, not its surface). The result is in meters per second. The orbital period — the time for one complete loop — follows from T = 2π r / v.

How to Use It

Enter the mass of the central body (for Earth, about 5.972 × 10^{24} kg) and the orbital radius. You can type scientific notation like 5.972e24. The calculator returns the velocity in m/s and km/s, plus the orbital period in seconds.

Worked Example

Consider a satellite orbiting at Earth's surface radius, r = 6.371 × 10^{6} m, with M = 5.972 × 10^{24} kg. Then

v = \sqrt{((6.674 × 10^{-11} × 5.972 × 10^{24}) ÷ (6.371 × 10^{6}))} ≈ \sqrt{6.256 × 10^{7}} ≈ 7,910\ m/s

or about 7.91 km/s. This is close to the well-known low-Earth-orbit speed of roughly 7.9 km/s.

使用注意事項與精準校對原則（6 大核心守則）

📐 1. 嚴格對齊輸入量綱與單位

輸入各項數值時請務必確認量綱（例如小時、天數、金額、百分比），避免單位不匹配引發推算偏差。

🔒 2. 純前端本機毫秒級隱私保障

所有計算邏輯 100% 在您的終端瀏覽器內完成，資料絕不上傳伺服器，嚴格守護商業與個人隱私。

📊 3. 符合公認學術與產業計算模型

演算法嚴格遵循國際通用數學公式與標準計算模型，經多重情境交叉核對，數值嚴密精確。

📋 4. 一鍵複製與免登入 PDF 匯出

支援一鍵複製推算結果字串至剪貼簿，亦可直接點擊列印按鈕輸出純淨 PDF 工作報表。

🔄 5. 多情景敏感度動態對比

建議針對極端值與多組預期情景分別試算，觀察輸出數據的波動區間，輔助做出更穩健的決策。

🛡️ 6. 專業諮詢與日常參考免責聲明

本工具運算結果供日常排程、學業研習與事前參考之用，如涉及重大法規或投資決策請諮詢持證專家。

常見問題 (FAQ)

Is the radius measured from the surface?

No — it is measured from the center of the central body. For a low orbit, add the body's radius to the altitude.

Does this assume a circular orbit?

Yes. This formula gives the speed for a perfectly circular orbit. Elliptical orbits have varying speed described by the vis-viva equation.

What about escape velocity?

Escape velocity is \(\sqrt{2}\) times the circular orbital velocity at the same radius.

🧮 orbital 速度與速率計算機