---
title: ASTR 511 W23 HW 4
author: Andy Tzanidakis and friends
tags: Galaxies Class
---

[TOC]

# Homework 4 - ASTR 511 W23
- Author: Anastasios (Andy) Tzanidakis
- Course ID: ASTR 511 W23

# Problem 1
## Part 1.1 - Install
> Install a Galactic Dynamics Package

!pip install gala 

We now have successfully downloaded Gala! :tada:

## Part 1.2 - 6D coordinates of Cluster
>To plot the orbit of your cluster you are going to need to know the full 6D phase space coordinates: RA, Dec, Distance, pm\_RA, pm\_Dec, Radial Velocity The first 5 coordinates should have been used in Homework 2 (and probably can be found easily on Simbad or in a paper via a quick Google or ADS search). Radial Velocity for the cluster (not for individual stars) will be needed to complete the coordinates! For many clusters this might be listed on Simbad. Or seriously just try Googling “Radial Velocity NGC XYZ”

For HW #2 I chose to study the open cluster NGC 188. Here are the following observed characteristics we derived for the cluster: 

We will use these parameters with gala - we note we will induce a [Plummer potential](https://ui.adsabs.harvard.edu/abs/1911MNRAS..71..460P/abstract) to our cluster given the derived mass from the isochrones:

c = coord.ICRS(ra=12.25839 * u.deg, dec=85.239882 * u.deg,
               distance=1310.552 * u.pc,
               pm_ra_cosdec=-8.649997442575426 * u.mas/u.yr,
               pm_dec=5.833591 * u.mas/u.yr,
               radial_velocity=-82.749 * u.km/u.s)

ngc_188_mass = 930 * u.Msun
ngc_pot = gp.PlummerPotential(m=ngc_188_mass, b=1*u.pc, units=galactic)

# Problem 2

## Part 2.1 - Create Tidal Stream
> This potential model needs to include at least a disk and halo component. If you’re using Gala you can (should) easily use the pre-packaged MilkyWayPotential, or build your own (useful if you want to make e.g. a heavier disk, etc). Does the cluster quickly disrupt (i.e in a few orbits?) How large does the stream get?

Okay now we will begin by creating a tidal stream of NGC 188. For this initial problem, we will assume a NFW potential for all three components of the Milky Way (i.e Disk, Halo and Dark Mater Halo). For the potential of the Disk we will assume 10$^{10}$ $M_{\odot}$, for the stellar Halo 10$^{10}$ $M_{\odot}$, and finally for the dark matter halo 10$^{12}$ $M_{\odot}$. We will forward evolve the cluster a few Myr to see if it survives -- we will be most interested in looking at the evolution in the Cartesian (X,Y,Z) space:

pot = gp.CCompositePotential()
pot['disk'] = gp.NFWPotential(m=7e10*u.Msun, r_s=40*u.kpc, units=galactic)
pot['halo'] = gp.NFWPotential(m=1e10*u.Msun, r_s=15*u.kpc, units=galactic)
pot['dm'] = gp.NFWPotential(m=1e12*u.Msun, r_s=15*u.kpc, units=galactic)

c_gc = c.transform_to(coord.Galactocentric).cartesian
ngc199_w0 = gd.PhaseSpacePosition(c_gc)
df = ms.FardalStreamDF()

gen_pal5 = ms.MockStreamGenerator(df, pot,
                                  progenitor_potential=pal5_pot)

pal5_stream, _ = gen_pal5.run(ngc199_w0, pal5_mass,
                              dt=200* u.Myr, n_steps=2_000, 
                              release_every=1, n_particles=1)

Here we show a movie starting from the current Cartesian coordinates and tracing its evolution forward in time for at least a few Myr. We find according to this simulation using a NFW potential that NGC 188 in a few million years will likely become a tidal stream. Interestingly, it take at least >100 Myr for the stream to fully develop, and closer to ~300 Myr the stream will likely will become fully phased mixed with the other stars in the Milky Way. I also found it very interesting that the cluster still remains relatively radially compact at ~10kpc from the Galactic center.

If we forward evolve it by 1Gyr - here we can see that it's fully phased mixed with the disk - in these cases it will likely be very hard to find the all the stars that started out from this open cluster

We can also see what the scale height will look like after a very long time. As we'd expect, the velocity dispersion will also increase the scale height of the disrupted stars. In this instance, we see that the remnants of NGC 188 will be something like ~1.5 kpc above the galactic plane.

We can very roughly trace what the dispersion in the Z direction will be as a function of time. After a few Myr the cluster will plateau in the scale height of the Galactic disk - this likely reflects something about the properties of the MW potential and the initial parameters of the cluster:

I would remiss not to mention that this exercise has made me appreciate how lucky we are in the Milky Way to have still dynamically surviving open and globular clusters. As we see from this exercise - most groups or collections of stars that orbit the MW potential will eventually be tidally disrupted making it very difficult to identify all members of of the cluster.

## Part 2.2 - NFW Potential
> Create a tidal stream using a large NFW potential Using the same initial 6D conditions as the MWY potential above, and hopefully a similar total Galaxy mass, try and generate the tidal steam as before with a simple NFW potential. Does the stream form? Does it form as quickly (i.e. in the same number or orbits)? Be sure to have some nice plots showing 2D or 3D projections of the orbit or stream. Some comparisons between the MWY and NFW results would be good. Tell me what you find!

Okay, now using our cluster we can think a little more about what would happen to the NGC 188 cluster if we change the mass profile of the Milky Way - for example, what if we didn't have a DM halo potential?

Let's begin with the hypothetical scenario of no DM potential. As we did in the previous problem, we will keep the same mass profile for the disk and halo, however, we will remove the DM component:

In the above figure we observe that a tidal stream of NGC 188 will occur, however, we notice right away that the scale height of the stars will be significantly higher compared the previous example. Also we notice some differences morphological features compared to the previous problem. For example, at the tails of the stream we notice that the dispersion of positions increases, also notice the axes. In the absence of a DM potential, the NGC 188 cluster will likely have a larger radial profile. This intuitively makes sense, if we remember that the NFW potential has a $\Phi(R) \approx -R^{-1}$, at larger radii the potential will slowly approach zero (see eqn. 1 [Naray 2009]([https://](https://arxiv.org/pdf/0810.5118.pdf))).

For example, here I have plotted the approximate potential functions for the NFW and Plummer potentials and we can see how for very large radii that the potential converges to zero - depending on which profile you use will dictate the steepness of that slope:

Now, what if we added back the DM potential, but now made it a less massive version. Here I adopt the mass of the DM halo to be $10^{10}$ while keeping the other component masses the same. In this example we see right away how the evolution of the NGC 188 cluster will differ. Now that we've added another potential we see that the the radial extent of the stream has decreased, but still not as compact it was in the initial problem. Additionally, with the presence of the DM potential we see how the morphological features of the cluster changes. For example, here we can see after 100 Myr that the cluster will likely be more clustered. My leading hypothesis (disclosure I don't understand that much about dynamics) is that with the presence of the DM halo, it is possible that clusters might have a larger timescale of disruption since the stars are likely experiencing forces from a much larger underlying potential:

Looking at the morphological distribution of streams can be a direct probe to the potential of the host galaxy. We can easily imagine how this exercise can be done for the local group galaxies, and see how clusters can disrupt under different galactic potentials.