Week 8 Exercises - PHYS 633 Astrophysical Dynamics

Part 1: Orbits in Disks¶

In this part of the exercise, we are going to investigate the orbit of a star in various potentials. The galpy documentation might be very helpful [https://docs.galpy.org/en/latest/index.html]. Using animations might help visualize how the orbits evolve instead of plots.

Start with a potential generated by a point mass. Use the KeplerPotential() function, and write a script which takes as inputs the position, the radial velocity, the tangential velocity, and the azimuthal angle $\phi$ . You can assume to be in the z=0 plane. Use $r_o=8$ kpc and $v_o=220$ km/s. What is the condition to be satisfied in order to have a circular orbit? Plot the rotation curve and the escape velocity curve. Discuss.
Now that you have a sense of how to use Orbit(), switch to an isochrone potential. Use the IsochronePotential() function, and write a script which takes as inputs the same parameters with the addition of the parameter ‘b’ passed into the potential function. How does the orbit evolve? Plot the same curves and discuss.
We can now move to an actual galactic potential! For exercises 3-5 use MWPotential2014() as the basis. Plot the level curves (you can use the code from the reading for this). Write a script which takes as inputs the energy, the z-direction angular momentum, the position. What do you notice? Plot the rotation and the escape velocity curves. Does it match your expectations? A reminder that this is how we can find the vertical velocity: $v_z=\sqrt{2(E-\Phi-\frac{v_R^2}{2}-\frac{L_z^2}{2R^2})}$
Now let’s add a super massive black hole of $M=4\cdot 10^6 M_{\odot}$ using a Kepler potential and do the same as 3). Can you notice a difference?
Finally, let’s tweak the dark matter content of the Milky Way on top of the SMBH. You can do so by applying the following transformation: MWPotential2014[2]*= 1.5 and do the same as 3). Is this a more accurate representation of the Milky Way by comparing your results with the literature? What’s the escape velocity required at the location of the solar system? What’s the lowest energy required to eject the Sun out of the Milky Way within 10 dynamical times?
Seperate the potential into a) the z-direction component b) the rotation plane. From the reading in section 9.2, find $R_{max}$ , $R_{min}$ , and $R_{median}$ curves. You can use toLinear() and toPlanar() methods for the orbit.

Part 2: Geometry of Resonances¶

Try to reproduce some of the panels in Figure 8.4 of Murray and Dermott.