math lessons - Standard assumptions in astrodynamics

For most of the problems in astrodynamics involving two bodies $m_1\,$ and $m_2\,$ standard assumptions are usually the following:

A1: $m_1\,$ and $m_2\,$ are the only objects in the universe and thus influence of other objects is disregarded,
A2: The orbiting body ( $m_2\,$ ) is far smaller than central body ( $m_1\,$ ), i.e.:

${m_2\over{m_1}}<<1$

Results:

A3: As the result of disparities in masses between $m_1\,$ and $m_2\,$ standard gravitational parameter ( $\mu\,$ ) includes only the mass of the central body, i.e.:

$\mu=G{m_1}\simeq{G}(m_1+m_2)$

A4: Orbit of orbiting body is not perturbed in any way, so the only orbits allowed are circular, elliptic, parabolic or hyperbolic.
A5: One focus of orbiting body's orbit coincides with the center of the central body,

The center of the central body can be taken as the origin of an inertial frame of reference for the orbiting body,

Examples where those assumptions do not hold

A1:
- although escape velocity is described as a velocity that should allow an orbiting body to coast to infinity and arrive there with zero velocity for most cases this will not be. E.g. even if the spacecraft is launched with escape velocity with respect to Earth it will not escape to infinity (e.g. leave the Solar system) because it will eventually succumb to the gravitational influence of the Sun.
- a rocket applying thrust
- in the case of atmospheric drag
A2: a binary star

If A2 is not fulfilled, many results still apply with a small modification, see the two-body problem in astrodynamics.