   # WMAP Effective Frequency Calculator Tool ALGORITHM DOCUMENTATION

## Introduction

The WMAP Effective Frequency Calculator is an interactive tool available on LAMBDA (see reference 1), which computes the effective frequency for a spectral energy distribution described by νβ in brightness temperature. The effective frequency is computed for the five WMAP frequency bands K, Ka, Q, V, and W, whose nominal frequencies are listed as 23, 33, 41, 61 and 94 GHz respectively. For those users wishing to embed the 9yr WMAP effective frequency computation in their code, we describe the underlying algorithm, which was originally presented in Jarosik et al. (2003), Section 7.4.1, with a small amendment for drift in the bandpass over the 9 year WMAP mission, discussed in Bennett et al. (2013) Appendix A.

## Algorithm

For a given value of β, the effective frequency over the full 9yr mission is computed for each of the five frequency bands. The computation can be broken into three main steps:

• (1) Compute the effective frequency per radiometer.

Table 11 of Jarosik et al. (2003) provides coefficients νavg, c1 and c2 from which the effective frequency ν0rad may be computed for each WMAP radiometer assuming a spectral index β of the diffuse sky component filling the beam:

ν0rad = νavg(1.0 + c1β + c2β2)(1/β),       (1)

where the frequencies are in GHz. Note that the above expression is not mathematically
defined for β = 0, which corresponds to the case of a source that has
a constant Raleigh-Jeans temperature. For cases in which |β| < 0.01, the tool
computes ν0rad from the average of the right-hand side of eqn 1 evaluated at two beta values bracketing 0.0 (β = ±0.01).

• (2) Average over radiometers to derive the effective frequency per band.

There are two radiometers for K-band (e.g., K11 and K12), two for Ka, four each for Q- and V-bands, and eight for W-band. For each of the five WMAP frequency bands, compute the flat-weighted average of ν0rad over the n radiometers in the band:

• (3) Apply the 9yr bandpass drift correction.

As described in Appendix A of Bennett et al. (2013), a small variation in the center frequency of K, Ka, Q and V bands was detected over the nine years of WMAP survey operations. The WMAP Effective Frequency Calaculator applies a small correction that accounts for this ‘bandpass drift’. The correction is a reduction of the pre-flight center frequencies (as computed in the two above steps) by 0.13, 0.12, 0.11, and 0.06% for K-, Ka-, Q-, and V-band, respectively; a W-band correction was not clearly detected and is set to zero:

δcorr = [0.13, 0.12, 0.11, 0.06, 0]/100.        (3)

And finally, the corrected effective frequency is

ν0corr = ν0 / (1. + δcorr)        (4)

## References

(1) WMAP Effective Frequency Calculator, effective_freq.cgi

(2) Jarosik et al. (2003) ApJS 145, p413.

(3) Bennett et al. (2013) ApJS 208, p20.

A service of the HEASARC and of the Astrophysics Science Division at NASA/GSFC  HEASARC Director: Dr. Andrew F. Ptak
LAMBDA Director: Dr. Thomas M. Essinger-Hileman
NASA Official: Dr. Thomas M. Essinger-Hileman
Web Curator: Mr. Michael R. Greason