Orbit Speed Inside and Outside a Mass Distribution

The orbit speed of an object is a measure of the amount and distribution of the mass that is attracting it gravitationally. To examine this dependence, consider a sphere of uniform density ρ which is diffuse enough to permit the orbiting of a mass m within the radius of the mass distribution. For this idealized case, the inverse square law form of the law of gravity along with the centripetal force relationship can be used to calculate the orbit velocity for a circular orbit. This calculation is made possible by the fact that the orbiting mass experiences a net attraction only by that mass inside its orbit, and the mass outside its orbit exerts a net zero force.

物体的轨道速率是衡量其引力吸引的物质总量和分布的量度。为了考察这种依赖关系，考虑一个密度均匀的球体ρ，该球体足够稀疏，允许在球体分布半径内轨道运行一个质量为m的物体。在这一理想化情况下，可以利用引力定律的平方反比形式以及向心力关系来计算圆轨道的轨道速率。这一计算得以实现的原因是，轨道上的物体仅受到其轨道内质量的净吸引力，而轨道外的质量对它施加的净力为零。

Upon calculating the circular orbit velocity, we find that inside the spherical mass distribution, the orbit velocity increases with radius, reaching a maximum at the radius of the mass distribution. Then if falls off as the radius increases, following the relationship:

在计算圆轨道速度时，我们发现，在球形质量分布内部，轨道速度随半径增加而增加，达到质量分布半径的最大值。然后，当半径继续增大时，速度会下降，遵循以下关系：

This calculation was done for a spherical mass, but for a uniform disc of mass like an idealized galaxy, the orbital velocity of material outside the detectable mass of the galaxy would be expected to fall off as some factor times the inverse square root of the radius. Surprisingly, the orbital velocity does not fall off at the edge of the detectable matter, staying almost constant well beyond the detectable edge of the galaxy. This is the nature of the evidence for dark matter which exceeds the visible matter by about a factor of six.

这个计算是针对一个球形质量进行的，但对于质量分布均匀的盘状物（如同一个理想化的星系），位于星系可探测质量之外的物质的轨道速度应随半径的平方根倒数成正比下降。令人惊讶的是，在可探测物质的边缘，轨道速度并未下降，而是在可探测边缘之外保持几乎不变。这是暗物质存在的证据，暗物质的质量大约是可见物质的六倍。