Redshift interval question

[Jared Coughlin]

Hello! I just have a quick question about the redshifts zA and zB. In equation 6 of the method paper, the limits of integration are from zA to zB. The paper defines the observer to sit at point B and the lightray to go from point A to point B. This means that the source of the light (the quasar, for instance) sits at point A and Trident is following the path of the light as it moves from the quasar at point A to the observer at point B. Also, the paper defines the redshift of the simulation output to be that of zA. 

In Hogg 1999 (the paper cited for equation 6), his limits of integration are from the redshift of the observer to the redshift of the source, which is the opposite of what equation 6 has. However, Hogg also implies that the redshift of the observer is less than the redshift of the source (that is, zB < zA). If this is the case, then the dz intervals given by equation 8 will be negative, since, as I understand it, dl is just the magnitude of the line segment, and therefore positive. This confuses me since this implies, from equation 9, that zi < zB. Doesn't that take you outside the bounds of the line of sight? That is, shouldn't zi > zB in order to "march towards" zA at the other end of the line of sight? If, however, zB > zA you still have the same problem, since dz will be positive and adding a bunch of positive numbers to zB will only make it larger, still taking you away from zA.  Unless there's an implied negative sign in l in equation 6 due to the limits being flipped?

So, my first question is whether or not zB < zA or not and which direction is defined to be positive?  

My second question also concerns that sum in equation 9.  The sum is from j = 1 to j = n, and the paper says that the lightray is broken into n segments, so that implies that the sum is over every segment? Wouldn't it need to only go from zB to the i'th segment? 

I don't think the paper is wrong, I'm just clearly misunderstanding something about how the sight line is being constructed, so any help understanding it would be greatly appreciated.  Thank you very much!

Sincerely,

-Jared 

[Cameron Hummels]

Hi Jared,

Thanks for the message.  The mathematical formalism can get a little bogged down in the paper, but it is a bit more clear in the code.  I'll respond inline below:

On Tue, Mar 20, 2018 at 7:42 AM, Jared Coughlin <Jared.W.C...@nd.edu> wrote:

Hello! I just have a quick question about the redshifts zA and zB. In equation 6 of the method paper, the limits of integration are from zA to zB. The paper defines the observer to sit at point B and the lightray to go from point A to point B. This means that the source of the light (the quasar, for instance) sits at point A and Trident is following the path of the light as it moves from the quasar at point A to the observer at point B. Also, the paper defines the redshift of the simulation output to be that of zA. 

That is correct.  We thought it was most intuitive to follow the path of the light from the absorber/emitter (point A, high z) to the observer (point B, low z).

 

In Hogg 1999 (the paper cited for equation 6), his limits of integration are from the redshift of the observer to the redshift of the source, which is the opposite of what equation 6 has. However, Hogg also implies that the redshift of the observer is less than the redshift of the source (that is, zB < zA). If this is the case, then the dz intervals given by equation 8 will be negative, since, as I understand it, dl is just the magnitude of the line segment, and therefore positive. This confuses me since this implies, from equation 9, that zi < zB. Doesn't that take you outside the bounds of the line of sight? That is, shouldn't zi > zB in order to "march towards" zA at the other end of the line of sight? If, however, zB > zA you still have the same problem, since dz will be positive and adding a bunch of positive numbers to zB will only make it larger, still taking you away from zA.  Unless there's an implied negative sign in l in equation 6 due to the limits being flipped?

 

So, my first question is whether or not zB < zA or not and which direction is defined to be positive?  

zB < zA.  If you generate a LightRay object, it is a series of arrays that are sorted from zA to zB, that is, from the absorber/emitter to the observer.  You can see this in action if you create a LightRay object for a dataset with redshift z:

>>> import yt

>>> import trident

>>> fn = <SOME DATASET>

>>> ds = yt.load(fn)

>>> ray_start = ds.domain_left_edge

>>> ray_end = ds.domain_right_edge

>>> line_list = ['H', 'C', 'N', 'O', 'Mg']

>>> ray = trident.make_simple_ray(ds, start_position=ray_start,

                              end_position=ray_end, data_filename='ray.h5',

                              lines=line_list)

>>> ad = ray.all_data()

>>> print(ad['redshift'])

This should print the redshift field for the LightRay object, which will start at z, the redshift of the simulation output, and then have minute decrements to slightly lower than z values as the ray travels from the source (zA, high z) to the observer (zB, low z).

As for the mathematical formalism in the paper goes, I reversed zA and zB in Hogg's equation (equation 6) as you noticed, which will yield negative values for comoving radial distance (i.e., l).  When you plug these negative values for l into equation 8, along with the negative values that come from (zB - zA), you get positive dz_i values.  Finally, plugging in the positive dz_i values into equation 9 and adding each one to zB (observer, low z), indeed gives you redshift values between zB and zA, the desired redshift range.  However, you are right, in that the i'th array element according to this equation is the i'th from the observer to the source, *not* the i'th from the source to the observer.  So in all truth, I think equation 9 should more correctly be: z_i = zA - sum().

 

My second question also concerns that sum in equation 9.  The sum is from j = 1 to j = n, and the paper says that the lightray is broken into n segments, so that implies that the sum is over every segment? Wouldn't it need to only go from zB to the i'th segment? 

Yes, the sum should go from j=1 to j=i to calculate the redshift of the corresponding i'th element of the array: z_i.  You can see how this is done if you look in the code directly: 

https://github.com/trident-project/trident/blob/master/trident/light_ray.py#L209

We've tried to do a good job of fully describing how to use the code, and what sort of datasets you'll get out of different functions in the docs and API docs at https://trident.readthedocs.io/en/latest/ , since they're built from docstrings directly in the code.  But you're right that we never explicitly describe the order of the arrays in the documentation.  I'll try to remember to add that to our FAQ.

 

I don't think the paper is wrong, I'm just clearly misunderstanding something about how the sight line is being constructed, so any help understanding it would be greatly appreciated.  Thank you very much!

 

Let me know if I didn't answer your questions regarding the code or if there are any other sources of confusion!

Cameron

--

Cameron Hummels

NSF Postdoctoral Fellow

Department of Astronomy

California Institute of Technology

http://chummels.org

[Jared Coughlin]

Hi Cameron,

Thank you! That was very helpful! I just have one more question, if you wouldn't mind? In equation 11, I'm assuming that v_i is the peculiar velocity of the gas and not the total velocity (peculiar + hubble)? Thanks!

-Jared

[Paryag Sharma]

Hi, I am also having problems with this redshift decreasing because then if we plot the spectrum using spectrum generator tool (for eg) Lyman alpha spectrum whose rest wavelength is 1216 it is blueshifting the dips because of decreasing redshifts how do I fix it ?