📚 Gravitational Fields and Satellites: Key Formula Derivations | 引力场与卫星：关键公式推导
Understanding gravitational fields and satellite motion is fundamental to A-Level Physics. This article derives the key formulas step by step, from Newton’s law of gravitation to escape velocity and Kepler’s third law. Each derivation is linked to core principles and exam requirements for OxfordAQA International A-Level Physics.
理解引力场和卫星运动是 A-Level 物理的基础。本文逐步推导关键公式，从牛顿万有引力定律到逃逸速度和开普勒第三定律，每个推导都紧扣核心原理和 OxfordAQA 国际 A-Level 物理考试要求。
1. Newton’s Law of Gravitation | 牛顿万有引力定律
Newton’s law of universal gravitation states that every point mass attracts every other point mass with a force directly proportional to the product of their masses and inversely proportional to the square of their separation.
牛顿万有引力定律指出，任何两个质点之间都存在引力，该力与它们的质量乘积成正比，与它们之间距离的平方成反比。
F = G M m / r²
The gravitational constant G = 6.67 × 10⁻¹¹ N m² kg⁻². This force is always attractive and acts along the line joining the centres of mass.
万有引力常数 G = 6.67 × 10⁻¹¹ N·m²·kg⁻²。该力始终为吸引力，作用线沿两物体质心的连线。
2. Gravitational Field Strength g | 引力场强度 g
Gravitational field strength g at a point is defined as the gravitational force per unit mass experienced by a small test mass placed at that point: g = F/m.
某点的引力场强度 g 定义为放置在该点的单位质量检验物体所受的引力：g = F/m。
For a point mass M, at a distance r from its centre, we derive g by substituting F = GMm/r² into the definition.
对于点质量 M，在距其中心 r 处，将 F = GMm/r² 代入定义式即可推导出 g。
g = G M / r²
g is a vector directed towards the centre of mass M, and its magnitude depends only on M and the distance r.
g 是矢量，方向指向质量 M 的中心，其大小只取决于 M 和距离 r。
3. Derivation of g for a Point Mass | 点质量 g 的推导
For a spherical body like a planet, the external gravitational field is the same as if all its mass were concentrated at the centre. Thus, at a distance r from the centre (r ≥ radius R), g = GM/r². On the planet’s surface, r = R, so the surface field strength is gₛ = GM/R².
对于像行星这样的球体，其外部引力场等同于全部质量集中于球心。因此，在距中心 r 处（r ≥ 半径 R），g = GM/r²。在行星表面，r = R，所以表面场强 gₛ = GM/R²。
Using Earth’s mass M = 5.97 × 10²⁴ kg and radius R = 6.37 × 10⁶ m, we obtain g ≈ 9.81 N/kg, which matches observed values. This derivation also explains how g decreases with altitude: at height h above the surface, g’ = GM/(R + h)².
代入地球质量 M = 5.97 × 10²⁴ kg，半径 R = 6.37 × 10⁶ m，可得 g ≈ 9.81 N/kg，与观测值一致。这一推导也解释了 g 如何随高度减小：地表上方高度 h 处，g’ = GM/(R + h)²。
4. Gravitational Potential V | 引力势 V
Gravitational potential V at a point is the work done per unit mass by an external agent in bringing a small test mass from infinity to that point without acceleration. Because the gravitational force is attractive, the work done is negative, and the potential is defined to be zero at infinity.
某点的引力势 V 是外部物体将单位质量从无穷远匀速移至该点所做的功。由于万有引力为吸引力，做功为负，且规定无穷远处势为零。
The force on a test mass m at distance x from M is F = GMm/x² directed radially inward. The work done from ∞ to r is W = ∫∞→r (GMm/x²) dx = -GMm/r. Dividing by m gives V.
在距离 M 为 x 处，检验质量受到的力为 F = GMm/x²，方向径向向内。从无穷远移至 r 所做的功为 W = ∫∞→r (GMm/x²) dx = -GMm/r。除以 m 即得 V。
V = -G M / r
The negative sign shows that potential decreases as one approaches the mass; an external agent must do positive work, but the field itself does negative work.
负号表明越靠近质量，势越低；外力需做正功，而引力场自身做负功。
5. Gravitational Potential Energy | 引力势能
The gravitational potential energy U of a system of two point masses M and m separated by distance r is obtained from U = mV.
两个相距 r 的点质量 M 和 m 所组成系统的引力势能可由 U = mV 得到。
U = -G M m / r
For a satellite, U is negative, indicating it is bound to the central body. As the orbital radius increases, U becomes less negative (increases), but kinetic energy changes accordingly – a topic we will explore next.
对于卫星，U 为负值，表明它被中心天体束缚。随着轨道半径增大，U 负值减小（能量增大），但动能也会相应变化——这将在后续内容中展开。
6. Circular Orbits and Satellite Motion | 圆轨道与卫星运动
A satellite in a circular orbit experiences a centripetal acceleration towards the centre of the planet. This centripetal force is provided entirely by the gravitational attraction between the planet and the satellite.
在圆轨道上运行的卫星具有指向行星中心的向心加速度。该向心力完全由行星与卫星之间的万有引力提供。
G M m / r² = m v² / r
Here M is the planet’s mass, m the satellite’s mass, r the orbital radius measured from the planet’s centre, and v the orbital speed. This equality is the starting point for deriving all circular orbit properties.
其中 M 为行星质量，m 为卫星质量，r 为从行星中心算起的轨道半径，v 为轨道速率。该等式是推导所有圆轨道性质的出发点。
7. Deriving Orbital Velocity | 推导轨道速度
By cancelling m and one power of r from the equation GMm/r² = mv²/r, we obtain v² = GM/r. Hence, the orbital velocity for a circular orbit is:
由方程 GMm/r² = mv²/r 约去 m 和一个 r，得 v² = GM/r。因此圆轨道的轨道速度为：
v = √(G M / r)
This result shows that v is independent of the satellite’s mass. For a given planet, satellites in lower orbits move faster. For example, a low Earth orbit at altitude ~200 km (r ≈ 6.57 × 10⁶ m) gives v ≈ 7.8 km/s.
这一结果表明 v 与卫星质量无关。对于同一颗行星，较低轨道上的卫星运动得更快。例如，高度约 200 km 的低地球轨道（r ≈ 6.57 × 10⁶ m），v ≈ 7.8 km/s。
8. Kepler’s Third Law from Newton’s Law | 从牛顿定律推导开普勒第三定律
The orbital period T is the time for one complete revolution. For a circular orbit, T = circumference / speed = 2πr / v. Substituting v = √(GM/r) yields T = 2πr / √(GM/r) = 2π √(r³/GM). Squaring both sides gives Kepler’s third law:
轨道周期 T 是运行一整圈所需的时间。对于圆轨道，T = 周长 / 速度 = 2πr / v。代入 v = √(GM/r)，得 T = 2πr / √(GM/r) = 2π √(r³/GM)。两边平方即得开普勒第三定律：
T² = (4π² / G M) r³
Although derived for a circle, this relationship also holds for elliptical orbits if r is replaced by the semi-major axis a. The constant of proportionality depends on the central mass M, so measuring T and r allows us to calculate M.
虽然由圆轨道导出，但这一关系在椭圆轨道中同样成立，只需将 r 替换为半长轴 a。比例常数取决于中心质量 M，因此通过测量 T 和 r 可以计算出 M。
This law is widely used in astronomy to determine the masses of planets, stars, and even galaxies.
该定律在天文学中被广泛用于测定行星、恒星乃至星系的质量。
9. Energy of an Orbiting Satellite | 轨道卫星的能量
The total mechanical energy E of a satellite in a circular orbit is the sum of kinetic energy K and potential energy U. Using v² = GM/r, we can express both in terms of r.
圆轨道卫星的总机械能 E 为动能 K 与势能 U 之和。利用 v² = GM/r，可将两者都用 r 表示。
K = ½ m v² = ½ m (GM/r) = GMm/(2r). Since U = -GMm/r, the total energy is:
K = ½ m v² = ½ m (GM/r) = GMm/(2r)。由于 U = -GMm/r，总能量为：
E = K + U = – G M m / (2 r)
E is negative, confirming the satellite is bound. Moreover, |E| = |U|/2. To move to a higher orbit (increase r), the total energy must become less negative, meaning energy must be supplied to the satellite, e.g., by firing thrusters.
E 为负，证实卫星被束缚。而且 |E| = |U|/2。要进入更高轨道（增大 r），总能量必须负值减小，即需要向卫星提供能量，例如通过发动机点火。
10. Escape Velocity Derivation | 逃逸速度推导
Escape velocity is the minimum speed an object must have at a planet’s surface to escape its gravitational field completely, arriving at infinity with zero kinetic energy. Use conservation of energy: initial K + U at surface must equal final K + U at infinity (both zero).
逃逸速度是物体在行星表面必须具有的最小速度，使其能够完全逃离引力场，到达无穷远处时动能为零。利用能量守恒：表面处的初始 K + U 必须等于无穷远处的 K + U（两者均为零）。
At surface (radius R): ½ m v_esc² + (-GMm/R) =
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