Contents

1 Introduction

1.1 Background summary

The spectral window at L-band (1.400-1.427 GHz) reserved for passive use only is important in microwave remote sensing for measuring parameters such as soil moisture and ocean salinity. At this frequency, radiation from celestial sources such as those in our galaxy can be significant and, unlike the constant cosmic background, spatially variable across the sky. A good description is needed for calibration (e.g. looking at cold sky) and to make corrections for the down-welling radiation that is reflected from the surface and detected by the sensor.

In addition to the cosmic background, the source of the radiation in the window at 1.4 GHz is line emission from neutral hydrogen and broadband (continuum) emission. Sources within our solar system, such as the sun, are not included. Radio astronomy measurements in this window have been made with modern instruments. This data has been converted into an equivalent thermal source with brightness

(1)

The equivalent brightness temperature T_b(Ω,ν) in the equation above has the characteristic that if it is integrated over the passband of a radiometer with a bandwidth of ΔB = 26MHz in the interval between 1.401 1.427 GHz, one obtains the same result one would obtain by integrating the actual data from the radio astronomy surveys. The conversion to passbands of other widths is described in the text. The reason for choosing the passband above is that it is representative of the Aquarius radiometer which motivated these studies [1]. In deriving the equivalent brightness temperature, it is assumed that the radiation is unpolarized. Section 2 below describes in more detail how the equivalent brightness temperature was obtained from the radio astronomy measurements.

1.2 Data

The data are presented as maps of the equivalent brightness temperature T_b(Ω,ν) in celestial coordinates. The data files are:

• TbGal_tot_CasAStockert.bin (binary): map of sky T_b with Cassiopeia A T_b spread over 1 pixel only,
• TbGal_tot_CasA1pix.bin (binary): map of sky T_b with Cassiopeia A T_b convolved by a Stockert-like beam,
• right_ascension.bin (binary): ECI J2000 right ascensions α (radian) for the maps,
• declination.bin (binary): ECI J2000 declination δ (radian) for the maps.

All files contain 721 × 1441 = 1038961 values, which is the number of elements for a map for declination between δ = -90^∘ and δ = +90^∘ at 0.25^∘ resolution and right ascension between α = 0^∘ and α = 360^∘ at 0.25^∘ resolution. All the data are double precision floating point (called ’double’ or ’float64’ in Matlab, ’real*8’ in Fortran or C).

This data will be listed on the website with this document

2 Derivation of the Equivalent Brightness Temperature

The brightness temperature of the sky background T_Sky is derived from the sum of three components: The brightness temperature of the continuum T_Cont (including, as a separate term, the effective emission of Cassiopeia T_{Cas A}^*), the effective brightness temperature of the atomic hydrogen T_HI^* and the cosmic microwave background T_CMB

The contribution from Cassiopeia A was excluded from the reference continuum survey, because of its strength and variability. We have added it back using another survey (see 2.3).

2.1 Continuum and CMB

The brightness temperature of the continuum plus CMB was measured by Reich and Reich [2, 3] for the northern sky (declinations larger than -20^∘), and by Testori et al. and Reich et al. [4, 5] for the southern sky. The measurements were performed at central frequency ν₀ = 1420 MHz with an effective bandwidth ΔB_C =18 MHz. The bandwidth spreaded over B_C = ν₀ ± 10MHz with an exclusion band of ±1MHz centered on ν₀ . The exclusion window is the location of the hydrogen line. The assumption is made that the continuum T_b is flat (independent of frequency) over the radiometer bandwidth. Hence the slight difference between the central frequencies of the sky survey and the radiometer (e.g. Aquarius radiometer) is neglected (see Appendix Appendix A and Appendix B), as well as the fact that T_Cont varies slightly over the band B (see Appendix Appendix A and Appendix C).

The continuum data are shown in Fig. 1 in clestial coordinates (ECI) for the epoch J2000. The data for the southern hemisphere were provided as a personal communication. The northern sky data are available for download at http://www.mpifr-bonn.mpg.de/survey.html). The data consist of a map of 721x1441 values of Tb, corresponding to a resolution of 0.25^∘ in both coordinates.

2.2 Atomic hydrogen HI

The HI line emission was measured by Hartmann and Burton [6] for the northern sky, and Arnal et al. [7] for the southern sky. Data were merged by Kalberla et al. [8] as an whole sky map, including a correction for stray radiation, at 891 different frequencies in the domain ν₀ ± 2.17MHz (equivalently: the measurements span between ±460 km/s Doppler velocities with a constant 1.03 km/s resolution). Contrary to the continuum, HI emission varies significantly with frequency even close to ν₀ , so that its T_b is not constant over the band B. Therefore, we define an effective brightness temperature T_HI^* derived from the integral (see Appendix Appendix D)

where T_HI (ν) is the T_b of HI at the frequency ν over the bandwidth dν, and is integrated over the band B_HI = ν₀ ± 1MHz that was excluded in the continuum survey ¹. Because B overlaps B_HI, the total flux of HI in the 2 MHz bandwidth has to be conserved when measured by the radiometer. Therefore, the integral is normalized by radiometer bandwidth, ΔB. One can verify that the flux emitted by HI in the band B_HI and the one measured by the instrument in the band B are the same.

Note: for another instrument measuring in the band B′ (with bandwidth ΔB′) overlapping B_HI, the map can be renormalized according to T_HI^*′ = (ΔB∕ΔB′)T_HI^*.

We use a sub-data set of 485 values limited to about ± 250 km/s Doppler shifts. The data are downloadable at http://cdsarc.u-strasbg.fr/ftp/cats/VIII/76/lab250.fit.gz The file contains a 3D map with a resolution of 0.5^∘ in galactic longitude and latitude, and about 1.03 km/s in velocity. The data are interpolated in the ECI J2000 reference frame at 0.25^∘ resolution in declination and right ascension (nearest neighbor) after the integration over the spectral band in (3) is performed for the data in the galactic reference frame.

2.3 Cassiopeia A

The location² of Cassiopeia A is in epoch J2000 [10], its diameter is of a few arcmin [9]. Its flux density varies with frequency and time. We use a flux density S_CasA = 1600 Jy at 1414 MHz for the period 2010-2013 (1 Jansky (Jy) is 10^-26 W/m²/Hz), as explained in Appendix Appendix E.

An effective T_b is derived, assuming it is homogeneous over a pixel in the map, according to

with Ω_p = sinδ(0.25^∘)² ≃ 1.629 × 10^-5 sr the solid angle of a pixel of size 0.25^∘× 0.25^∘ at a declination δ. Figure 3 reports the location of Cassiopeia A in the sky and Fig. 5 reports its brightness temperature when arbitrarily spread over one pixel.

Another approach to report Cassiopeia A on the map is to simulate how it could appear when observed by the same instrument used for the Sky continuum survey. For this purpose, the signal from Cassiopeia A is convolved with a gaussian beam of aperture ~ 35′ and derive the T_b according to

2.4 Total emission map

Figure 6 shows the total emission. It is the sum of the continuum, atomic hydrogen, cosmic background and Cassiopeia component described previously, all assumed unpolarized (see Appendix Appendix F). The map assumes a bandwidth ΔB = 26 MHz. The data are available as described in Section 1.2.

Acknowledgment

We would like to thank P. Reich and W. Reich for their help with the continuum survey data and Peter Kalberla for his support with the HI data.

Appendix A Spectral index of continuum emission and Cassiopeia A at decimeter wavelengths

The dependence of the flux density S_ν of a source on frequency ν is characterized by the spectral index α_S such that

In the literature, the spectral index refers also to the dependence of the T_b on ν. As T_b ∝ S_νν^-2, the spectral index β_S for the T_b is defined by

and is related to α_S by β_S = 2 -α_S. Note that for a black body, β_S = 0 (the T_b is constant with the frequency) and α_S = 2 (the flux density increases as the square of the frequency in the Rayleigh-Jeans domain).

For the sky continuum, α_S is in the range [11] or equivalently β_S is in the range . For Cassiopeia A, the spectral index varies in time. For the years 2010-2013, we use α_S = -0.727 (see Appendix Appendix E).

Appendix B Impact of the difference between central frequencies of Sky surveys and radiometer center frequency

The sky survey data were acquired at a central frequency slightly different than the one at which Aquarius and other salinity remote sensing radiometers operate. In this section, the impact of this slight difference is quantified.

Define the relative difference between the galactic T_b at the frequency ν = 1420 MHz and the one at a given frequency ν_m used for salinity measurement as

For a T_b spectral index β_S, the brightness temperature at a frequency ν (in MHz) writes

(9)

and the relative difference simplifies to

Results computed for the central frequency (ν_m = 1414 MHz) at various spectral indices representative of the sky are reported in table 1. The relative difference between the T_b at the central frequencies of the survey and ν_m is of the order 1%.

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βS 2.4 2.7 3.0 R1@1414 MHz +1.021% +1.150% +1.278% Table 1: Relative difference between brightness temperatures at the frequency of 1420 MHz used for the continuum measurements and the frequency used for salinity measurements νm assuming various spectral indices βS. Here νm = ν = 1414 MHz (Aquarius).

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Appendix C Impact of changes in continuum Sky T_b inside the bandwidth of the radiometer

Typically, a radiometer integrates over a finite bandwidth. Over this bandwidth, the sky T_b is expected to vary slightly. This variation is quantified in this section following two approaches:

1. one quantifies the difference between the sky’s T_b at the center of the band and at its bound, and
2. one quantifies the difference between the sky’s T_b at the center of the band and the effective T_b derived from the flux integrated over the band.

For the first approach, similarly to (8), define the relative difference between the sky T_b at the central frequency ν_m where the measurements are performed and the one at a frequency bound ν_lim as

or, assuming a spectral index β_S,

Results computed for the band limits (ν_lim = 1401 MHz and ν_lim = 1427 MHz) are reported in table 2. The sky T_b at the bounds of the measurements frequency band differs from the one at the center of the band by a few percent.

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βS 2.4 2.7 3.0 1401 MHz +2.241% +2.525% +2.810% 1427 MHz -2.172% -2.441% -2.708% Table 2: Relative difference between brightness temperatures at the frequency used for salinity measurements νm and the boudaries of the frequency band assuming various spectral indices βS. Here νm = ν and the boundaries frequencies are 1401 MHz and 1427 MHz.

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For the second approach, one defines the relative difference between the effective T_b over the radiometer bandwidth and the T_b at the center frequency. Define the effective brightness temperature T_eff as the temperature of a black body which would radiate the same power over the band as the source does:

This leads to

(15)

Replacing λ by c∕ν, one obtains

(16)

Assuming a temperature spectral index of β_S, the brightness temperature at a frequency ν becomes

(17)

and the effective temperature becomes

with

(20)

For β_S ≠ 3,

(21)

and after rearranging, one finally obtains the ratio between the effective temperature and the T_b at ν as

(22)

For β_S = 3,

(23)

Then, one defines the relative difference in T_b as

The results for the radiometer in this example are reported in table 3 and show a very small difference between the sky T_b at the central frequency and the effective T_b associated with the brightness over the whole measuring band.

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βS 2.4 2.7 3.0 1414; 1401; 1427 (MHz) -2.10-3 % -1.10-3 % +1.10-7 % Table 3: relative differences between effective brightness temperatures integrated over the Aquarius frequency band B with various spectral indices βS and the brightness temperature at the Aquarius central frequency ν.

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Appendix D Effective brightness temperature in a small bandwidth

The brightness B over a bandwidth Δν = (ν₀ + Δν∕2) - (ν₀ - Δν∕2) = ν₁ -ν₂ is derived integrating the spectral brightness, function of T_b(ν), according to

Define the effective brightness temperature T_b^* so that a source with the brightness temperature T_b^* constant over Δν would have the brightness B in (26). Therefore, one wants

Extracting T_b^* from the integral in (27) and using (26) for B, on derive

If one assume 1∕λ² varying little over Δν (with Δν = 2MHz and ν₀ = 1420MHz, 1∕λ² varies by ~ 0.3%) so that it can be extracted from the integrals in (30), one derive the simple expression

Appendix E Flux density of Cassiopeia A at 1.4 GHz in the period 2010-2013

Cassiopeia A flux density varies with frequency and time [12, 13]. Therefore, to derive the flux density relevant to Aquarius, one must transform existing measurement to the proper frequency (1.414 GHz) and time (2010 - 2013). Parker reports historical measurements performed between the years 1955 and 1967 and the frequencies 10 MHz and 15.5 GHz. He adopts a temporal decrease for the flux density of 1.1±0.18% per year [12]. Baars et al. [13] report measurements performed between 1959 and 1973 and 10.05 MHz and 31.41 GHz. They propose the following law for the decrease in flux density

(32)

where the frequency ν is in GHz and dS∕S is the relative decrease in flux density per year (note: log is the base 10 logarithm). At a frequency ν = 1.414 GHz, the decrease is about 0.92%, a value relatively close to the one chosen by Parker.

The raw data reported by Parker and Baars et al. is presented in Fig. 7. Both data sets are in relative good agreement, particularly in the high frequency domain. These data are expressed at the epoch in 1965, as well as in 2010 that is our epoch of interest (Fig. 8) using (32). The transformation to epoch 1965 leads small changes only, but the transformation to 2010 induces a significant decrease of flux density at all frequencies and a noticeable change of spectral shape. We derive linear regressions of the spectral data for frequencies larger than 300 MHz using both data set separated (see Fig. 8 legend for results). For merged data sets, we find the following spectral function for the year 2010

(33)

which leads to a flux density at 1414 MHz S_CasA(1414) = 1578 Jy. For the year 2013, (32) leads to S_CasA (1414) = 1534 Jy.

Appendix F Polarization of the Sky background

The background map (Fig. 6) presented here assumes that the signal is unpolarized. Data on the degree of polarization at L-band has recently been published. The large-scale polarization of the Sky background was measured in the vicinity of ν₀ by Wolleben et al. [14] for the Northern sky and Testori et al. [15] for the Southern sky. The first survey was conducted at a central frequency of 1410 MHz with a bandwidth of 12 MHz (10 Mhz for some of the measurements). That sets the upper bound of the frequency band at 1416 MHz, less than the lower bound of the HI emission band. The latter survey was conducted at a central frequency of 1435 MHz, with a bandwidth of 14 MHz, avoiding the HI band too. Claimed accuracies are ~15 mK r.m.s. noise,  less than 50 mK systematic error and a pointing accuracy of the order of the arc minute.

Appendix F.1 Merging the data sets for northern and southern sky.

3rd Stokes, northern sky 4th Stokes, northern sky 3rd Stokes, southern sky 4th Stokes, southern sky Figure 9: Map of (left column) the third Stokes parameter and (right column) the forth Stokes parameter, from (top row) the northern sky survey and (bottom row) the southern sky survey. The colorscale is arbitrarily saturated.

3rd Stokes 4th Stokes Figure 10: (top row) Map and (bottom row) histogram of the differences between the southern sky map and the northern sky maps (see Fig. 9) over the overlapping region.

3rd Stokes 4th Stokes Figure 11: Scatterplot of the data from the northern and southern sky survey. The green line represent a linear regression through the data and the red dashed line has a slope of 1. The correlation coefficients are 0.83918 and 0.8272 for the third and forth Stokes parameters respectively.

Maps for polarization are available for the northern (DRAO, [14]) and southern sky seperatly (Villa Elisa, [15]), but no merged product exists to our knowledge. The goal of this section is to describe the method used to merge the two maps into one, and to perform some basic sanity check on the data. Both maps were downloaded at the address http://www.mpifr-bonn.mpg.de/survey.html in J2000 RA/DEC coordinates at the original resolution (i.e. 0.25^∘). The map from both survey are reported in Fig. 9. They overlap each other for declinations between -10^∘ and -29^∘ .

The differences between the two map over the overlapping region are reported in Fig. 10. A scatterplot of the data from both survey is reported in Fig. 11.

In order to merge the surveys, the data over the overlapping region are averaged between the two maps.

Appendix F.2 Analysis of the polarization of the Sky background

(a) (b) Figure 12: Maps of (a) the third Stokes parameter in K and (b) the fourth Stokes parameter in K [14, 15]. The colorscale has been arbitrarily bounded at ±0.45 K.

Figure 12 reports the map of the polarized Sky. The general structure of the map is very different from the emission maps in figures 1 and 2. The Galactic plane does not exhibit a particularly large polarization, and is characterized by large structures with a dynamic range of the order of 2K.

(a) (b) Figure 13: Histograms of (a) the third Stokes parameter and (b) the fourth Stokes parameter.

Figure 13 reports the histogram for the polarized maps in figure 12. The polarization translates in third and fourth Stokes mostly between -0.5K and +0.5K, with maxima up to ±2 K. The polarized is ignored in the final map.

Appendix G Glossary

ν

central frequency for Aquarius measurements................................................... 37

ΔB

bandwidth for the Aquarius measurements.......................................................4

B

band of frequencies for Aquarius measurements.................................................. 5

T_Sky

brightness temperature of the sky background ................................................... 5

T_HI ^*

effective brightness temperature of the atomic hydrogen..........................................5

T_{Cas A} ^*

effective brightness temperature of Cassiopeia A................................................. 5

T_CMB

brightness temperature for the cosmic microwave background.................................... 5

ΔB_C

bandwidth for the measurement of the continuum emission.......................................5

B_C

band of frequencies for the measurement of the continuum emission.............................. 5

α

right ascension...................................................................................5

δ

declination ...................................................................................... 5

S_ν

flux density at frequency ν ..................................................................... 25

α_S

spectral index for the flux density .............................................................. 25

β_S

spectral index for the brightness temperature................................................... 25

References