More Applications#
There are a lot of other processes in astrophysics that are described by a system of ODEs. Here are some suggestions to explore.
Trojan asteroids in the Sun-Jupiter system.
The trojan asteroids share the same orbit as Jupiter around the Sun, but lead or trail it in its orbit by 60 degrees. These can have interesting orbits around the L4 and L5 Lagrange points. This is also an example of the restricted 3-body problem. To make the integration easier, we work in the rotating frame defined by Jupiter’s orbital period. Then Jupiter is stationary and we integrate the orbits of the Trojan asteroids. Initial conditions for some interesting orbits have been worked out by Rabe 1961—see table I. This topic is discussed extensively in the text by Paul Hellings.
A one zone model of X-ray bursts.
X-ray bursts are the thermonuclear explosion of a thin accreted layer on the surface of a neutron star. The dynamics of this burning can be approximated as a toy model looking at the heating and the depth of the fuel layer, as described in Heger et al. 2018 and Paczynski 1983 This simple system can give rise to repeated bursts (and a limit cycle ) depending on the accretion rate.
Star formation in the Galaxy.
The paper by Bodifee 1986 works out a simple model for star formation in the Galaxy, tracking the gas and stellar mass history. This can show limit cycle behavior, as explored in Sharaf et al. 2012. This topic is also discussed extensively in the text by Paul Hellings.
General relativistic orbits:
Geodesics around a Schwarzschild black hole
The motion of a test particle in general relativity around a black hole of mass \(M\) described by the Schwarzschild metric leads to an ordinary differential equation. See the Schwardzschild geodesics Wikipedia article for references.
Precession of Mercury
The orbit of Mercury precesses due to general relativistic effects. This is nicely worked out in Körber et al. 2018.