Models of Halo Emission

Written Report

This is the first project that you will do as a group. Your group will hand in a single report with all group members contributing and listed as authors. The report should read as a single, complete study on the given problem. The report should be similar to a scientific paper with sections for Introduction and statement of the problem (a few sentences each), description of the models and code (turn in the code separately, do not include it in the body of the report), results/predictions from the models for given parameters, and a discussion of the results. It should address all the questions asked in the assignment description.

The beta model

Read the paper "Constraining the Milky Way's Hot Gas Halo with O VII and O VIII Emission Lines" by Matthew Miller and Joel Bregman at http://adsabs.harvard.edu/abs/2015ApJ...800...14M.
To begin, we need an analytical model for the hot gas density in the halo. We chose a model with a small number of parameters and will eventually compare the model predictions with the observations to constrain the parameters. We define the halo model in terms of coordinates relative to the Galactic center. The location of an emitting region is specified in terms of R, the radius in the disk, and z, the height above the disk for a flattened halo model or in terms of r, the distance from the Galactic center (galactocentric radius), for a spherically symmetric model.
The beta model is defined in terms of the beta parameter, which gives the dependence of density on radial position, and the normalization that sets of the scale of density. For a spherically symmetric model there is one additional parameter, the core radius (r_c), while for a flatten disk model there are two additional parameters, the core radii for R and z. Write down the definitions of the beta model for the two cases.
In addition to the density, our model of the halo must also specify the temperatures of the gas. What do Miller and Bregman assume for the distribution of temperatures in the halo? What is the observational basis for their assumption? Following Miller and Bregman's temperature assumption and using your results from the previous project assignment, find the emissivity for O VII and O VIII. Compare with the values quoted by Miller and Bregman.
For this assignment, do not consider the local bubble. That will be the topic of the next assignment.

Calculating the line intensities

The halo model is defined in coordinates referenced to the Galactic center, but our observations are made from near Earth and are specified by the direction on the sky, set by (l, b) in Galactic coordinates, and the distance along the line of sight, s. Write down equations to calculate the galactocentric radius, r, from l, b, and s. Specify any fixed values use in the equations.
We will assume that the halo is optically thin. This means that we neglect the effects of absorption and scattering. Therefore, to find the total line intensity we need to integrate over the line of sight.
From the AtomDB website, the flux observed at Earth (with no redshift or absorption) for a single temperature plasma is
where epsilon(T_e) is the emissivity in ph cm^3/s, R is the distance to the source in cm, and the integral over N_e N_H dV is the emission measure in cm^-3. Note that this assumes one is measuring the flux from an entire object that fits well within the field of view of the telescope and that the size of the object is small compared to its distance.
Show how to convert this formula to one appropriate for the halo. Assume that N_e = N_H = n. Note that you will need to move R inside the integral since we observe different parts of the halo at different distance. Note that the units for line intensity are photons s^-1 cm^-2 str^-1. For any line of sight, we observe a fixed fraction of the sky set by the field of view of the instrument (in this case given in steradians). This corresponds to different physical sizes at different distances. (It may be useful to draw or think of a cone.)
Write Python code to evaluate your integral over the line of sight distance for a specific choice of l, b and beta model and parameters. Write code for both the spherically symmetric beta model and the flattened disk model. When writing the integration code, there are several things to keep in mind and discuss in your report. For example, what are the bounds of integration? Is the result sensitive to those bounds? Is the integrand mathematically well behaved over the full range of integration? Note also that any numerical integration of an continuous function is necessarily an approximation. How accurate is your integration?
This page is useful for doing numerical integration with Python: http://docs.scipy.org/doc/scipy/reference/tutorial/integrate.html.

Using line intensity maps to distinguish between different models

We will now use your line intensity calculation code to illustrate how the line intensity varies across the sky in different models and what sorts of measurements would be useful in constraining models. Note that measurements along the Galactic plane are subject to high levels of absorption, so typically observations at high Galactic latitudes are used to constrain the properties of the halo.
For Galactic latitudes of b = +30°, +45°, and +65°, plot total line intensity for OVII plus OVIII versus Galactic longitude for two models: 1) a spherically symmetric model with a core radius of 3.5 kpc, and 2) a flattened disk model with R_c = 3.5 kpc and z_c = 0.5 kpc. Use beta = 0.5 for both models. Put both curves for each Galactic latitude onto the same plot for ease of comparison.
For Galactic longitudes of l = 90°, 180°, and 270°, plot total line intensity for OVII plus OVIII versus Galactic latitude for same two models. Put both curves for each Galactic longitude onto the same plot for ease of comparison.
Discuss the results. How can one distinguish a spherically symmetric model from a flattened disk?
We will now examine the effect of the beta parameter limiting ourselves to the a spherically symmetric model with a core radius of 3.5 kpc. Use models with beta = 0.4, 0.6, 0.8, and 1.0. Make line intensity plots of your choosing. How can one distinguish between different values of beta?
In your report, discuss any resulting thoughts that you have regarding this work.