Differential equations of satellite motion

Satellite orbit differential equation ：

   The central force in satellite motion is a conservative force , The law of conservation of mechanical energy holds . therefore , We can use the total energy to discuss the specific form of satellite orbit . In the inverse square gravity problem , force The specific form of is known .

   Because intentional force is conservative force , So there must be potential energy , Yes

  (2) In style It is a quantity that has nothing to do with satellites but only with the earth , For the distance between the satellite and the earth , For satellite quality . Take the potential energy at infinity as zero , Then the distance between the particle and the force center is The gravitational potential energy is ：

   When we study the problem of mental force , Pictured 1 Shown , The law of conservation of mechanical energy and the law of conservation of moment of momentum are usually combined . The following two equations are commonly used as basic equations ：

chart 1· Establish polar coordinate system for analysis

   In the formula Is constant    To unite and Type available ：

   In order to eliminate Type in the , Let's do the following transformation first ：

   take Into the Available ：

   take Jointly available ：

   figure out , And separate the variables and integrate to get ：

   The standard conic equation of the known polar coordinate system is ：

   Make ,, So you can put the formula Rewrite into The form of the formula . By the type You know , because The value of is always positive , So the total energy Determines the shape of the satellite orbit .

   Let's discuss it in categories ：

, be , The orbit is an ellipse ;

, be , The orbit is a parabola ;

, be , The orbit is hyperbolic .

Numerical simulation

1、 Circular orbit

chart 2· Numerical simulation of circular orbit

chart 3· The energy change of the satellite in circular orbit

2、 Elliptical orbit

chart 4· Numerical simulation of elliptic orbit

chart 5· The energy change of satellite in elliptical orbit

3、 Parabolic orbit

chart 6· Numerical simulation of parabolic orbit

chart 7· The energy change of the satellite in parabolic orbit

3、 Hyperbolic orbit

chart 8· Numerical simulation of hyperbolic orbit

chart 9· The energy change of satellite in hyperbolic orbit

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