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WMAP

Wilkinson Microwave Anisotropy Probe

Product Name
Beam Maps
Mission
WMAP
Coord. System
Focal Plane coordinates
Projection Type
Rectilinear, pixelized at 2.4 arcminutes (0.04°)
Resolution
0.23°- 0.93° (frequency dependent)

Description

The main and near-sidelobe response of each of the 20 WMAP antenna feeds has been mapped in-flight using observations of Jupiter. The 9-year release comprises 17 Jupiter observing seasons:

Oct/Nov 2001
Feb/Mar 2002
Nov/Dec 2002
Mar/Apr 2003
Apr/May 2004
Jan/Feb 2005
Dec 2003/Jan 2004
May/Jun 2005
Feb/Mar 2006
Jun/Jul 2006
Mar/Apr 2007
Jul/Aug 2007
Apr/May 2008
Aug/Sep 2008
May/Jun 2009
Oct/Nov 2009
Jul/Aug 2010
As a prelude to beam analysis, an archive of calibrated time-ordered observations is constructed, consisting of Jupiter passages within roughly 7.0, 5.5, 5.0, 4.0 and 3.5 degrees of either the A- or B-side beam center for K, Ka, Q, V and W bands respectively. The time-ordered observations are corrected to a fiducial Jupiter distance of 5.2 AU, background subtracted and corrected for aberration. To constrain low signal-to-noise beam pedestals, a hybrid TOD archive is then constructed in which model predictions (Bennett, et al., 2013; Hill, et al. 2008) are substituted for data at the 2,3,5,6 and 9 dBi levels of K,Ka,Q,V and W respectively. This hybrid beam archive serves as the basis for beam map and window function analysis.

For purposes of constructing beam maps, the data in the hybrid beam TOD archive are assigned to 2.4 arcminute bins on a coordinate grid centered on either the A or B-side focal plane axis. The beam response for each feed is computed from the average temperature in each bin. No correction has been made for the side-A vs. side-B input transmission imbalance. These beam maps are convenient for some applications, but are not used in the computation of the flight beam transfer functions. The 2.4 arcminute binning acts as a smoothing kernel which filters high frequency spatial content. The pixelization transfer function may be estimated from the Legendre transform of the symmetrized radial profile of the binning kernel. Assuming a square pixel of 0.04 degrees on a side centered on the origin, the symmetrized radial profile of the binning function may be represented as

f(r) = 1.0for r < R

= 1.0 - (4/pi)*acos(R/r)for R <= r <= R*sqrt2
where R = 0.02 deg. Both the pixelization profile and pixelization transfer function are provided as a useful reminder of the limitations of the 2.4 arcminute binning.

Beam maps are provided in 10 FITS image format files, one file for each differencing assembly. Each file contains:

  • the beam map for the A side, in mK (antenna temperature)
  • the statistical error of each bin of the A side beam map, in mK (antenna temperature). The statistical error is based on the number of observations in each bin. Model points are assigned 100% error.
  • the beam map for the B side, in mK (antenna temperature)
  • the statistical error of each pixel of the B side beam map, in mK (antenna temperature) Model points are assigned 100% error.

The beam coordinates form an equal area rectangular coordinate system centered on the optic axis of the spacecraft. They are related to coordinates theta (elevation from optic axis) and phi (azimuth about optic axis) as follows:

  • Xbeam = 2*sin(theta/2) * cos(phi)
  • Ybeam = 2*sin(theta/2) * sin(phi)

The "optic axis" of the spacecraft is elevated by 19.5 degrees from the S/C XY plane and lies within the S/C YZ plane. Although this vector is close to the S/C Y axis (+ or - depending on A or B side), it becomes the Z axis of the focal plane coordinate system.

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Product Name
Beam Radial Profiles
Mission
WMAP

Description

For each differencing assembly, an azimuthally symmetrized radial beam profile is computed by binning the ensemble of individual A- and B- side hybridized Jupiter observations. A constant bin size of 0.25 arcmin is used, and the straight mean of all hybrid samples within a radial bin represents the value for that bin.

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Product Name
Beam Transfer Functions
Mission
WMAP

Description

Beam transfer functions are computed from the Legendre transform of the binned hybrid radial beam profile. The window function applicable to power spectra is the square of the beam transfer function.

Beam transfer functions are presented as ASCII tables, with the first column being multipole moment l and the second column the transfer function b_l (amplitude) normalized to 1.0 at l=1. A third column has been added for the 9-year release, which contains the fractional 1-sigma error, delta_b_l/b_l. The method by which the error is computed is described in Hill et al. 2009, ApJ 180, 246.

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Product Name
Beam Transfer Function Fractional Covariance Matrices
Mission
WMAP

Description

A full l-by-l covariance matrix is provided for each WMAP beam transfer function, by DA, as discussed in Hinshaw et al. 2007, ApJS 170, 288. An efficiently-compressed version of this information is contained in the WMAP likelihood code, using the procedure discussed in Appendix A of the above paper. In the notation of Appendix A2 of Hinshaw et al. (2007), Bll′ = <ul ul′>, where ul = Δ(bl)/bl is the fractional error in the beam transfer function bl.

The matrices are provided in FITS image format, as a separate file for each DA. This is a symmetric matrix whose rows and columns are implicitly indexed by ascending multipole moment; the range of l is 0 to lmax, where lmax corresponds to NAXIS1 - 1 and is DA-dependent. Since bl is normalized to 1.0 at l=1, Bll′ is zero for entries with this multipole value. The square root of the diagonal of the matrix is identical to the fractional 1-σ error provided in the Beam Transfer Function product. The method by which the fractional covariance is determined is described in Hill et al. 2009, ApJ 180, 246.

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Additional Information

A service of the HEASARC and of the Astrophysics Science Division at NASA/GSFC
Goddard Space Flight Center, National Aeronautics and Space Administration
HEASARC Director: Dr. Alan P. Smale
LAMBDA Director: Dr. Eric R. Switzer
NASA Official: Dr. Eric R. Switzer
Web Curator: Mr. Michael R. Greason