## Sunday, 1 June 2008

### Gravitational Potential

Last night I was playing with Mathematica, visualising perturbations to the gravitational potential of one body by the presence of another (e.g. for binary stars, star-planet and planet-moon systems). I did this by writing a gravitational potential function, using
V(r)=\frac{G M}{r}

Actually, I allowed my Mathematica function to take arguments beyond just a distance r... Firstly I split r into x and y, using the relationship
r^2=x^2+y^2
, then allowed the mass M to be parameterised as well as an offset from the origin, since
r^2=(x-a)^2+(y-b)^2
describes a circle of radius r centred on (a,b).

Finally, I used the fact that potentials can be added to generate the true potential of interacting bodies, i.e.
V_\mathrm{total}=V_\mathrm{star}+V_\mathrm{planet}
. With these functions defined, and a few basic quantities such as the mass of the sun (2*10^30 kg) and the distance in metres corresponding to 1 AU (149,598,000,000 metres), I was able to use the Mathematica ContourPlot[] function to plot equipotential lines for various two-body potentials. I was looking for particularly interesting perturbations to the otherwise circularly symmetric (spherically, in 3D) potential of a "point mass". The approximation of a point mass works perfectly as long as we're dealing only with space outside of the true radius of the object, and on the length scales we're looking at here, that is definitely the case.

I've included some of the more interesting equipotential plots below, including the parameters used to plot them. Remember, you can click any of the images to see a bigger version.

Potential around the Sun: MSun at (0,0) on a scale from -10AU to +10AU on both x and y axes

Earth-Sun System: MSun at (0,0) and MEarth at (1AU,0). x-axis from 0.8 AU to 1.2 AU, y-axis from -1 AU to 1 AU

Earth-Moon System: MEarth at (0,0) and MMoon at (384000km,0). Both axes from -2 to +2 times the Earth-Moon separation (384,000 km)

Sun-Jupiter System: MSun at(0,0), MJupiter at (5.15 AU,0). Axes x from 4 AU to 6 AU, y from -2 AU to +2 AU

HD209458
One of the most "famous" exoplanetary systems, HD209458b is a Hot Jupiter orbiting the star HD209458 (note that the 'b' added to the star name gives the planet name, i.e. the planet is the second object (discovered) in the system). HD209458 has a mass of 1.01 MSun, and the planet HD209458b has mass around 0.69 MJupiter. The separation of the two bodies is 0.045 AU.
Axes from 0.02 AU to 0.07 AU on the x, -0.05 AU to +0.05 AU on the y

HD41004B
Another exoplanet system, this time HD41004B is 0.4 MSun and the planet, HD41004Bb is 18.4 MJupiter. The separation is 0.0177 AU and the axes run from -0.03 AU to +0.03 AU on both x and y

Cygnus X
Cygnus X is a 30 solar mass star and an 8.3 solar mass black hole. The separation is 0.2 AU. I plotted this one with the star at -Separation/2, the black hole at Separation/2 in the range -0.25 AU to +0.25 AU on both x and y axes

PSR B1913+16
A binary pulsar (the first one detected, I think). A 1.441 Solar mass neutron star and a 1.387 Solar mass companion star. Separation 0.0130356021 AU. Axes from -0.02 AU to +0.02 AU on both x and y

I also tried plotting some 3D equipotential surfaces, but the results weren't quite so impressive. Here's the equipotential surface for the Cygnus X system in 3D, range -0.5 AU to 0.5 AU in all three dimensions:

After this, I tried to visualise some of the general relativistic potentials (i.e. corrections to the standard Newtonian potential to take into account general relativistic effects) but didn't get any of them to produce nice visualisations, yet!