📚 Orbiting the Earth | 绕地球轨道运行
When a satellite orbits the Earth, it is continuously falling towards the planet due to gravity but moving forward fast enough to miss it. This balance between gravitational pull and tangential velocity results in a stable orbit, governed by Newton's law of gravitation and circular motion principles.
当卫星绕地球运行，它因重力不断向地球坠落，但同时又具备足够的切向速度使得它能绕过地球。万有引力与切向速度之间的平衡决定了稳定的轨道，这由牛顿万有引力定律和圆周运动原理支配。
1. Gravitational Force as Centripetal Force | 万有引力作为向心力
For a satellite of mass m orbiting the Earth (mass Mₑ) at a distance r from the centre, the gravitational force provides the necessary centripetal force for circular motion:
对于质量为m的卫星，在距地心r处绕地球（质量为Mₑ）运行，万有引力提供圆周运动所需的向心力：
GMₑm / r² = mv² / r
where G is the gravitational constant and v is the orbital speed. The satellite's mass m cancels out, indicating that the orbital speed is independent of the satellite's mass.
其中G是万有引力常数，v是轨道速度。卫星质量m被消去，表明轨道速度与卫星质量无关。
2. Orbital Speed Equation | 轨道速度方程
By rearranging the centripetal relationship, we obtain the orbital speed for a circular orbit:
通过重新整理向心力关系，我们得到圆轨道的轨道速度：
v = √GMₑ / r
eases as the orbital radius increases. For a low Earth orbit (e.g., r ≈ 6800 km from Earth' relationship ensures that satellites closer to Earth move faster than those farther away.
这表明轨道速度随轨道半径增大而减小。对于近地轨道（如距地心约6800 km），速度约为7.6 km/s。这种反比关系保证了离地球越近的卫星运行越快。
3. Orbital Period and Kepler's Third Law | 轨道周期与开普勒第三定律
lete revolution. Since the circumference is 2πr, T = 2πr / v. Substituting v yields Kepler's third law:
圆轨道的周期T是完成一整圈所需的时间。周长为2πr，T = 2πr / v。代入v得到开普勒第三定律：
T² = (4π² / GMₑ) r³
Thus, the square of the period is proportional to the cube of the orbital radius. This applies to any satellite around Earth and even to planets around the Sun, with the central mass Mₑ replaced by the Sun's mass. Knowing the period allows calculation of orbital radius, which is essential for satellite positioning.
因此，周期的平方与轨道半径的三次方成正比。这适用于任何绕地球运行的卫星，也适用于绕太阳的行星，只需将中心质量Mₑ换成太阳质量。知道周期就能计算轨道半径，这对卫星定位至关重要。
4. Energy of an Orbiting Satellite | 轨道卫星的能量
al potential energy U = -GMₑm/r (assuming zero at infinity). Therefore, total energy E = Eₖ + U = -GMₑm/(2r).
圆轨道卫星的总机械能（动能+势能）是常量。动能Eₖ = ½ mv² = GMₑm/(2r)。引力势能U = -GMₑm/r（以无穷远处为零势能）。因此总能量E = Eₖ + U = -GMₑm/(2r)。
The negative total energy indicates that the satellite is bound to the Earth. To move to a higher orbit, energy must be supplied (e.g., by firing thrusters); to drop to a lower orbit, energy must be removed (e.g., through atmospheric drag). The magnitude of the kinetic energy is half the absolute value of the potential energy in a circular orbit.
负的总能量表明卫星被地球束缚。要转移到更高轨道，需要提供能量（例如通过点火推进器）；要降到更低轨道，则需要移除能量（例如通过大气阻力）。在圆轨道上，动能的大小是势能绝对值的一半。
5. Geostationary Orbits | 地球同步轨道
, remaining fixed relative to a point on the surface. Using T = 24 h (86,400 s) and Kepler's third law, the required orbital radius is approximately 42,200 km from Earth's centre, giving an altitude of about 35,800 km.
地球同步轨道卫星在地球赤道上空运行，周期正好为24小时，相对于地面某点保持静止。利用T = 24 h（86,400 s）和开普勒第三定律，所需的轨道半径约距地心42,200 km，高度约35,800 km。
The orbit must be equatorial and in the same direction as Earth's rotation (west to east). These satellites are crucial for communications, television broadcasting, and weather monitoring because ground antennas can point to a fixed position in the sky.
轨道必须在赤道面内，且运行方向与地球自转同向（自西向东）。这类卫星对通信、电视转播和气象监测至关重要，因为地面天线可以指向天空中的一个固定位置。
6. Weightlessness in Orbit | 轨道上的失重现象
Astronauts in orbit experience apparent weightlessness not because gravity is absent, but because both the astronaut and the spacecraft are in free fall towards Earth, accelerating at the same rate. There is no contact force to give a sensation of weight.
轨道上的宇航员体验到失重，并不是因为重力消失，而是因为宇航员和航天器都在向地球自由下落，加速度相同。不存在接触力，因此没有体重的感觉。
Thus, the condition is equivalent to being in a continuously falling lift. True weight (gravitational force) is still acting and provides the centripetal force needed to keep them in a curved path. A weighing scale would read zero because it falls with the astronaut.
因此，这种状态等同于处于不断下落的电梯中。真正的重量（万有引力）仍然作用，并提供了保持弯曲路径所需的向心力。体重计会读数为零，因为它随宇航员一起下落。
7. Orbital Transfers and Hohmann Transfer | 轨道转移与霍曼转移
To change from one circular orbit to another, a satellite performs orbital manoeuvres. A Hohmann transfer orbit uses an elliptical path that is tangent to both the initial and final orbits, minimising energy use.
要从一个圆轨道变到另一个圆轨道，卫星需要进行轨道机动。霍曼转移轨道利用一个椭圆路径，与初始和最终轨道都相切，从而使能量消耗最小。
Two engine burns are required: one to enter the elliptical transfer orbit, and another to circularise at the new radius. This principle is used to send satellites to higher orbits or for interplanetary missions. The time taken for transfer is half the period of the ellipse.
需要两次发动机点火：一次进入椭圆转移轨道，另一次在新的半径处圆化。这一原理用于将卫星送入更高轨道或进行行星际任务。转移所需的时间是椭圆轨道周期的一半。
8. Orbital Speed vs Escape Velocity | 轨道速度与逃逸速度
For a given distance r from Earth's centre, the circular orbital speed is v = √GMₑ/r, and the escape velocity is vₑ = √2GMₑ/r. Escape velocity is √2 times the orbital speed. At the Earth's surface, escape velocity is about 11.2 km/s, while low Earth orbital speed is about 7.9 km/s.
在距地心r处，圆轨道速度v = √GMₑ/r，逃逸速度为vₑ = √2GMₑ/r。逃逸速度是轨道速度的√2倍。在地球表面，逃逸速度约为11.2 km/s，而近地轨道速度约为7.9 km/s。
If a satellite gains enough kinetic energy to make total mechanical energy zero, it can escape Earth's gravitational field. This requires reaching the local escape velocity. A rocket leaving Earth must exceed 11.2 km/s to become unbound.
如果卫星获得足够的动能使总机械能为零，它就能脱离地球引力场。这要求达到当地的逃逸速度。离开地球的火箭必须超过11.2 km/s才能成为无束缚状态。
9. Factors Affecting Real Orbits | 影响实际轨道的因素
Real satellites experience small perturbations: atmospheric drag at low altitudes causes orbital decay, lowering the satellite's altitude over time; the Earth's equatorial bulge creates precession of the orbital plane; solar radiation pressure and gravitational influences of the Moon and Sun also alter orbits.
实际卫星会受到小的扰动：低轨道的大气阻力导致轨道衰减，随着时间的推移降低卫星高度；地球赤道隆起造成轨道平面进动；太阳光压以及月球和太阳的引力也会改变轨道。
To maintain a stable orbit, satellites often carry thrusters for station-keeping, especially geostationary satellites which must remain precisely positioned above a specific longitude. Low-Earth-orbit satellites have limited lifetimes due to drag unless they are periodically boosted.
为了保持轨道稳定，卫星通常配备推进器进行位置保持，尤其是必须精确定位于特定经度上方的地球同步卫星。近地轨道卫星由于阻力寿命有限，除非定期进行轨道抬升。
10. Summary of Key Equations | 关键公式总结
The following table summarises the main relationships for a satellite of mass m orbiting Earth (mass Mₑ) at radius r (circular orbits):
下表总结了质量为m的卫星绕地球（质量Mₑ）在半径r处运行的主要关系（圆轨道）：
| Quantity | Expression |
|---|---|
| Orbital speed v | v = √GMₑ / r |
| Orbital period T | T² = (4π²/GMₑ) r³ |
| Kinetic energy Eₖ | Eₖ = GMₑm / (2r) |
| Potential energy U | U = -GMₑm / r |
| Total energy E | E = -GMₑm / (2r) |
| Escape velocity vₑ | vₑ = √2GMₑ / r |
These expressions assume a point-mass Earth and circular paths. For elliptical orbits, the semi-major axis a replaces r in Kepler's third law, and energy considerations differ.
这些表达式假设质点地球和圆路径。对于椭圆轨道，在开普勒第三定律中半长轴a替换r，能量考虑也有所不同。
Published by TutorHao | Physics Revision Series | aleveler.com
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