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 ν^{β} 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 ν_{avg}, c_{1} and c_{2} from
which the effective frequency ν_{0}^{rad} may be computed for each WMAP radiometer
assuming a spectral index β of the diffuse sky component filling the beam:

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 ν_{0}^{rad} 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 ν_{0}^{rad} over the n radiometers in the band:

ν_{0}^{band} = Σ (ν_{0}^{rad})i/n, (2)

(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: