10–13 Mar 2014
Berlin (Adlershof)
Europe/Berlin timezone

Tutorials - Wednesday

Today, we want to calculate for our toy model source the line-of-sight integral
 
 , usually expressed in units of   
 
respectively the astrophysical factor
 
  , usually expressed in units of    ,
 
 
numerically with CLUMPY. (Remark: as a homework exercise for preparation or consolidation for today's tutorial, you can perform these calculations also analytically by hand. See bonus exercise below)

In a second exercise, we will use this result together with yesterday's result to calculate the flux of photons per time and detector area from our toy model source.

Recall the properties of our toy model:

Constant density rho_0 = 150 GeV/cm^3 within a sphere of
radius R = 0.2 kpc at
distance d = 8.5 kpc from the observer.


Task 0: Units conversion


Usually, the astrophysical factor and line-of-sight integral are calculated in units of kiloparsecs (for distances) and sun masses   (for masses). Verify:
 
 
and express the density  of our toy model source in these units. What is the interesting fact when translating  into (or vice versa)?
 

Task 1: Numerical integration of and J with CLUMPY


1a. Make sure that our toy model clump results from the Einasto profile    for the Einasto profile parameter  , with   and  .
 
 
1b. Perform the line-of-sight integrations with CLUMPY
 
In the folder where executing CLUMPY, you need the parameter files toytarget_params.txt and toytarget.txt. In the file toytarget_params.txt, the general parameters for a clumpy simulation are set. This file also calls the file toytarget.txt, where our specific toy target is defined. At the very end of the toytarget_params.txt file, the numerical precision and the resolution (gSIMU_ALPHAINT_DEG) of the two dimensional grid on the celestial sphere are set.
 
Then you can let CLUMPY calculate a 2D J-factor skymap of the source (h5 mode) with the command:
 
~/CLUMPY_v2011.09.CPC_corr3/bin/clumpy -h5 toytarget_params.txt toytarget 4 1
 
The value "4" gives the size of the field of view in degrees, the value "1" is related the the numerical accuracy of the simulation and should not be changed.
 
There is a python script, HAPworkshop.toytarget.py, in the output folder for integrating the total J-factor from the 2D-skymap output by CLUMPY (and plotting the CLUMPY 2D-skymap output data using the healpix package). You can execute it by
 
python HAPworkshop.toytarget.py -w 4 -a 0.05 -v cart
 
By parsing the size of the field of view (-w 4) and the resolution (-a 0.05; the value set for gSIMU_ALPHAINT_DEG) of the CLUMPY simulation to the script, it should correctly find the corresponding CLUMPY output file. The option -v can be set to cart (cartesian plot) or moll (mollweide projection of the whole celestial sphere) to select the desired way of plotting. With
 
python HAPworkshop.toytarget.py -w 4 -a 0.05 -v cart -r 0.5
 
you can then fold the skymap with the detector resolution given in degrees.

Then try also the h2 mode of CLUMPY:
 
~/CLUMPY_v2011.09.CPC_corr3/bin/clumpy -h2 toytarget_params.txt 0.01 2 10 0 0
 
Compare the value of the total integrated J-factor from the h2 mode calculation, written in the file toytarget.txt.Jalphaint.output, with both the result from the 2D skymap and the analytic result, given by:

    (analytic result)

Increase the numerical precison in the toytarget_params.txt file and redo the h2 mode calculation. (For the 2D skymap/h5 mode, you have to wait a long time for the results when increasing the precision...).

 
1c. Optional: Play with CLUMPY and more realistic profiles, e.g., the Einasto profile with   or the Zhao family, parsed to CLUMPY by kZHAO instead of kEINASTO in the toytarget.txt file. For example, the Navarro-Frank-White profile is given by  kZHAO with #1 = 1, #2 = 3, #3 = 1.
 

Task 2: Calculate fluxes


Now fix as photon spectrum the process of dark matter annihilating into up or down (anti-)quarks, given by:
 
 
with    ,  , and the parameters:
 
a = -1.5
b = 0.047
c = -8.7
d = 9.14
e = -10.3
 
2a. Calculate and plot  , using the total astrophysical factor calculated above.
 
2b. Set the threshold energy of our detector to   and calculate the flux Phi of photons per time and detector area from our toy model source.

Hint for 2a and 2b: Remember our "Master formula" (for Majorana Dark Matter particles):
 


 
2c. Compare your calculated flux with the flux from Crab nebula above 100 GeV.

Hint: This publication may be useful: http://arxiv.org/abs/1110.2987


Bonus Task 3: Calculate the astrophysical factor for our toy model analytically:



(Note also the comments given by David Maurin in this pdf)

3a. For our spherically symmetric toy model, the line-of-sight integral depends only on the angle  between the line of sight (l.o.s.) and the line connecting the center of the sphere and the observer on Earth. What is the maximum   beyond which   will be 0?
 
 
3b. Now calculate the line-of-sight integral   as a function of   and plot the result (this is the function for the "analytic solution"-curve in the above python script profile-plot!).
 
Hint: Use the law of cosines for an arbitrary triangle with sides a, b, c and angle  opposite to side a,
 
 ,
 
and make geometrically sure that for given a,b, and  there are two drawable triangles, i.e., solutions for the side length c.
 
 
3c. Now perform the integration over the  solid angle   and calculate J as a function of  , i.e., integrate from   to   .
 
Hints:
  • In spherical coordinates   with   , the line element  reads:  .
  • Locate the connecting line between the center of the sphere and the observer onto the z-axis, i.e.   in these coordinates
  • Make sure that for our toy model source the approximation  is well fulfilled and simplify the integral correspondingly.
  • Then use: 
(Comment for highly motivated people: The integration is also solvable (e.g., with the help of Mathematica) without assuming the above approximation for  )
 
 
3d. Plot the result of 1c (as a function of  ). What result do we finally get for the total astrophysical factor when integrating over the full source, i.e., setting   ? (The result is the value for the comparision of the CLUMPY output with the analytic result!)
 
 
3e. Convince yourself, that the above integrations become much more difficult (and analytically inaccessible), when taking a non constant density (plug in, e.g.,  , with r the radial distance from the toy model's center.