Bilingual edition: English is preserved and Chinese follows each unit. Terminology uses the confirmed v20260916 glossary; automated semantic review remains traceable.
中英双语版：英文原文完整保留，中文紧随对应单元；术语采用 v20260916 确认表，自动语义审校结果可追溯。

Circular Orbit

圆轨道

Gravity supplies the necessary centripetal force to hold a satellite in orbit about the earth. The circular orbit is a special case since orbits are generally ellipses, or hyperbolas in the case of objects which are merely deflected by the planet's gravity but not captured. Setting the gravity force from the universal law of gravity equal to the required centripetal force yields the description of the orbit. The orbit can be expressed in terms of the acceleration of gravity at the orbit.

重力提供了使卫星保持绕地球轨道运动所需的向心力。圆轨道是特殊情况，因为轨道通常为椭圆，或者在物体仅被行星重力偏转但未被捕获的情况下为双曲线。将由万有引力定律得出的重力力设等于所需的向心力，即可描述轨道。轨道可以表示为轨道上重力加速度的函数。

The force of gravity in keeping an object in circular motion is an example of centripetal force. Since it acts always perpendicular to the motion, gravity does not do work on the orbiting object if it is in a circular orbit.

保持物体在圆周运动中的重力是向心力的一个例子。由于它始终垂直于运动，如果物体在圆轨道上运动，重力不会对物体做功。

Calculation

计算
 Index

Gravity concepts

Orbit concepts
索引 重力概念 轨道概念
 
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Earth's Gravity

重力

The weight of an object is given by W=mg, the force of gravity, which comes from the law of gravity at the surface of the Earth in the inverse square law form:

物体的重量由 W=mg 给出，这是地球表面的万有引力定律所体现的反平方律形式的引力作用力：

At standard sea level, the acceleration of gravity has the value g = 9.8 m/s2, but that value diminishes according to the inverse square law at greater distances from the earth. The value of g at any given height, say the height of an orbit, can be calculated from the above expression.

Above the earth's surface at a height of h =m = x 106 m, which corresponds to a radius r = x earth radius, the acceleration of gravity is g = m/s2 = x g on the earth's surface.

Please note that the above calculation gives the correct value for the acceleration of gravity only for positive values of h, i.e., for points outside the Earth. If you drilled a hole through the center of the Earth, the acceleration of gravity would decrease with the radius on the way to the center of the Earth. If the Earth were of uniform density (which it is not!), the acceleration of gravity would decrease linearly to half the surface value of g at half the radius of the Earth and approach zero as you approached the center of the Earth.

请注意，上述计算仅在h为正值时（即在地球外部的点）给出正确的重力加速度值。如果你在地球中心钻一个孔，那么在接近地球中心的过程中，重力加速度会随着半径的减小而减小。如果地球具有均匀的密度（实际上它并非如此），那么重力加速度会线性减小，到地球半径的一半时，重力加速度会减至地表值g的一半，并在接近地球中心时趋于零。
Hole through center of Earth
穿过地球中心的空洞
在标准海平面，重力加速度的值为 g = 9.8 m/s²，但根据距离地球更远的逆平方定律，该值会减小。在给定高度（例如轨道高度），可以利用上述表达式计算重力加速度。在地球表面以上高度为 h = 10⁶ m，即对应半径 r = 1 地球半径的位置，重力加速度为 g = 9.8 m/s² = 1 地球表面的重力加速度。
中文译文中的待填/计算数值依次对应：1：h 2：h6 3：rr 4：gp 5：gr。实际数值以上方原输入框为准。
 Index

Gravity concepts

Orbit concepts
索引 重力概念 轨道概念
 
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Binary Circular Orbit

二体圆轨道
From the gravity force and the necessary centripetal force:
If you are riding on one of the masses, the relative motion equation has the same form if you substitute the reduced mass
which gives the orbit equation:
从重力和所需的向心力来看：如果你在其中一个质量上骑行，相对运动方程在代入约化质量后形式相同，从而得到轨道方程：

This leads to Kepler's 3rd law (the Law of Periods) which is useful for the analysis of the orbits of moons and binary stars.

这导致了开普勒第三定律（周期定律），该定律在分析卫星轨道和双星系统轨道时很有用。

Since the period T of the orbit is given by

then the motion equation can be written
which reduces to
由于轨道的周期T已给出，因此可以将运动方程写成，这可以简化为

If we use the convenient astronomical units

  • r = a in AU (astronomical units)
    r = a（以天文单位为单位）
  • G = 4π2
    m in
    m in（专名或术语）
    		 (solar masses)
    太阳质量
  • T in years
    年份中的 T
then this takes the form
如果我们使用方便的天文单位，那么它就具有以下形式
This applies to circular orbits where
a is the radius, and to elliptical orbits
where a is the semi-major axis.
这适用于圆轨道，其中a是半径，也适用于椭圆轨道，其中a是半长轴。
and from just the period and orbit radius you can obtain the sum of the masses m1 + m2. If you can obtain the individual orbit radii r1 and r2 then you can use the center of mass
质心
condition

with the measured mass sum to obtain the individual masses
These relationships are important in the study of visual binaries
视觉双星
. 		Index

Gravity concepts

Orbit concepts
索引 重力概念 轨道概念
 
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Reduced Mass

约化质量
The relative motion of two objects that are acted upon by a central force can be described by Newton's 2nd Law as if they were a single mass with a value called the "reduced mass".

两个受中心力作用的物体的相对运动可以由牛顿第二定律描述，如同它们是一个具有称为‘约化质量’的单个质量一样。

From Newton's 3rd Law:

从牛顿第三定律：

The relative acceleration of the two masses is

两个质量的相对加速度是

Since a2 is negative, this can be rewritten in terms of the magnitudes of the quantities:

由于 a² 为负数，这可以改写为量的大小：
Applications:
应用：
Binary orbit
二体轨道
 Rotation of molecule
分子的旋转
H-D doublet
H-D 双重线
 Index

Gravity concepts
索引 重力概念
 
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仅通过周期和轨道半径，就可以得到两质量之和 m + m。如果你还能获得各自的轨道半径 r 和 r ，就可以利用测得的质量之和条件来求得各自的质量。这些关系在研究中是重要的。