meeting 2026 04 01 gw - JacobPilawa/TriaxSchwarzschild_wiki_6 GitHub Wiki

Context

  • Doing some sanity checks regarding the K-band K-corrections here. I think that the most direct answer to "are we implementing the K-corrections correctly" is answered with Equation 3 and Figure 8 from this paper which explicitly relates the absolute magnitude to the K-correction. I have more information on this in the "Diagnostics" section below, but the long-story short is that equation 3 says:
M = m - mu - K(z)

  • And then their plot of K(z) is negative for z>0, so the absolute magntiudes become more positive (fainter) with the k-correction assuming m and mu are fixed.

  • I also have the exact equations as they enter my code below, which appear to be consistent with the paper above.

  • Lastly, I threw together one final "simulation" using a galaxy template too so that we can look at what a real galaxy spectrum looks like at rest vs. redshifted and how that compares to the K-band filter bandpass.

Quick information related to email exchange

  • One thing I wanted to mention is that I think one of the statements in the email exchange is maybe a cause of confusion:
I think everyone agrees on this definition, where I use C to mean K-correction. 

m = M + mu + C   or    M = m - mu - C

And a positive K-correction  (i.e. C >0) means the galaxy appears fainter.

  • I'm interpreting things in this way -- we are trying to compute the absolute K-band magnitude M since this how we compute the mass, so we should focus on the right hand equation (effectively fixing m) rather than the left hand equation (effectively fixing M).

  • If the K-correction is positive, the expression for M = m - mu - C means that M becomes more negative, which is intrinsically brighter. That is to say, if we have a galaxy with m = 15, mu = 40, and C = 0.2, we would get:

    • M = 15 - 40 = -25.0 without a K-correction
    • M = 15 - 40 - 0.2 = -25.2 WITH the k-correction which makes the absolute magniude brighter.

  • If the k-correction is negative, though -- say C=-0.2, we get:

    • M = 15 - 40 = -25.0 without a K-correction
    • M = 15 - 40 - (-0.2) = -24.8 WITH a K-correction, which makes the absolute mangnitude fainter.

How it's actually computed in my code

  • My code computes the absolute K-band magnitude with this line:
df['Mk'] = kcorr_to_abs_mag(kcorr_vals, z)
  • The function "kcorr_to_abs_mag" takes in kcorr_vals which are the k-corrected magnitudes as well as the redshift and returns kcorr_vals - DM, where DM is the distance modulus computed as:
d_L = cosmo.luminosity_distance(z).to(u.pc).value # converts z --> distance in pc
DM = 5 * np.log10(d_L / 10.0) # distance modulus
  • kcorr_vals is computed as:
kcorr_vals = kband_magntiudes - kcorr_Kband(z)
  • where kcorr_Kband is exactly the function defined in Chilingrian+10 and kband_magnitudes are from the 2MPZ catalog. Put into one equation, this is:
kcorrection k band = (raw kband) - kcorrection - DM
  • which is the same as teh above expression.

worked example

  • I picked a random row from 2MPZ which has the following:

    • z = 0.218246
    • observed k band = 13.803

  • The absolute magnitude is thus:

    • M = m - k_correction(z) - DM(z)
      • m = 13.803
      • k_correction(z) = -0.3899
      • DM = 40.169
    • This gives M = 13.803 - (-0.3899) - 40.169 = -25.9761 WITH k-correction
    • This gives M = 13.803 - 40.169 = -26.366 WITHOUT k-correction
    • Meaning that the k-correction in my code is making the absolute magnitude larger, which means FAINTER.

applied to the full catalog

  • Here's what the above looks like (without the distance modulus term) applied to all fo 2MPZ:
2MPZ Full
images/260401/applied.png

Diagnostics

  • I started by trying to find references that explictly spell out their apparent --> absolute magntiude calculation and managed to find An imaging K-band survey – II. The redshift survey and galaxy evolution in the infrared which constructs a K-band luminosity function with ~127 galaxies.

  • In their section 5.2, they state:

  • "In order to obatin absolute magnitudes, we require knowledge of the luminosity distance D, the K-band k-correction K(z), and the aperture correction A(Z)":

M(z) = m - 5\log_{10}(D_L/10pc) - K(z) + A(z)
  • Note that this is the same convention as used in Hogg+99 (the A(z) aperture correction is unique to their analysis, we can ignore it), just rearranged (equation below is how Hogg+99 has it written):
m = M + 5\log_{10}(D_L/10pc) + K
  • They then include a plot of their k-corrections, as well as two K(z) equations (one equation is the version they derive, the other is from Rocca-Volmerange & Guideroni, 1988):
K(z) = \frac{-2.58z + 6.67z^2 - 5.73z^3 -0.42z^4}{(1 - 2.36z + 3.82z^2 -3.53z^3 + 3.35z^4)} [Glazebrook+95]

K(z) = -(1+(5z)^{(-3/2)})^{(-2/3)} [RVG]
Plot from Glazebrook+95 Plate from Manucci+01 Glazebrook+95 vs. Chilingarian and RVG
[images/260401/k.png]]](/JacobPilawa/TriaxSchwarzschild_wiki_6/wiki/[[images/260401/man.png) images/260401/all_k_corrs.png
  • This interpretation appears to be consistent with what Manucci+01 says as well. They plot K(z) - K(0), and what we're interested in is K(0) [the rest frame K band magnitude]. His plot is negative for all values of z, suggesting that K(z) < K(0). This is a statement about magntiudes, so K(z) is brighter than K(0). That is to say, the rest frame K band should be fainter than the observed K-band.

A small example with a galaxy template

  • As an additional sanity check, I have worked through an example where I take a sample galaxy spectrum from the SWIRE library, and I redshift it to z=0.5 and have some of the various bands marked. We can see that the rest frame spectrum has lower flux at the K-band part of the spectrum than the version which is redshifted to z=0.5. That is to say -- the true intrinsic brightness is smaller than the observed brightness, and so the k-corrections should make galaxies fainter:
Example Spectrum
images/260401/example.png

Chilingarian Calculator

KCorrection Calculator
images/260116/kcorr.png
  • The Chilingarian calculator that they themselves wrote give negative k-corrections which, when applied using Hogg's equation above (since the K correction term has a negative sign attached when computing absolute magnitude), results in fainter galaxies.