Uranus Brightness Calibration

Aim

To measure Uranus' brightness temperature in order to use it as an accurate flux calibrator.

Introduction

Uranus is used by ATCA users as a primary flux calibrator for observations at frequencies between 16 GHz and 110 GHz. It is used as a flux calibrator because it was thought that its brightness temperature was a constant, save for the changing distance between it and the Earth, which can be accounted for.

However, when trying to calibrate 1934-638 (the ATCA primary flux calibrator for low frequency observations) at high frequencies, using Uranus caused the flux of 1934-638 to appear significantly different at 12mm wavelengths than it did before when Bob Sault made his original measurements at those wavelengths. Since we assume that the flux of 1934-638 does not change over time (which has proven to be the case at cm wavelengths), we tried to understand what was varying.

Mark Wieringa found evidence that the Uranus brightness temperature changes as the planet's orientation to us changes (http://dx.doi.org/10.1016/j.icarus.2006.04.012). This is due to the polar regions of Uranus having a different temperature to the rest of the planet's atmosphere. We have not yet found a model that correctly accounts for this orientation effect, so the best chance we have is to periodically measure Uranus' brightness temperature so it can again be used as a reliable flux calibrator.

Background

we need a model that takes into account the orientation of Uranus

Kramer, Moreno and Greve (2008, http://adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2008A%26A...482..359K&db_key=AST&link_type=ABSTRACT) talk about Uranus' variability at ~90 GHz and how viewing angle affects the apparent brightness of Uranus.

Some things of note:

  • In 1992, Uranus had a TB of 136 K at 90 GHz.
  • Uranus is slightly oblate, with a polar radius of 24973 km and an equatorial radius of 25559 km.
  • In 1985, Uranus was showing its South Pole, with a sub-earth point (SEP) latitude of -82 degress, while in 2005 it showed its equator, with an SEP latitude of -6 degrees. Uranus' equinox will occur in 2007.
    • the sub-earth point is the point on the planet from which the Earth would be observed to be directly overhead
  • Uranus has a maximum angular diameter of 4.084 arcsecs (=1.979859793E-5 rads), when the equatorial radius is presented to Earth at the minimum Earth-Uranus distance of 2581.9E6 km
    • So for all our recommended mm frequencies:
Frequency (MHz) Wavelength (m) Maximum unresolved baseline (m) In klambda
17000 0.017634850 890 50.508
19000 0.015778550 796 50.508
33000 0.009084620 458 50.508
35000 0.008565499 432 50.508
44000 0.006813465 344 50.508
46000 0.006517227 329 50.508
93000 0.003223575 162 50.508
95000 0.003155710 159 50.508
  • Any flux scaling (with mfboot for example) should limit to 50 kilolambda if possible, so as to be sure of containing all the Uranus flux within the synthesised beam.
    • this can be achieved with "select=uvrange(0,50)" in miriad
  • Kramer et al. (2008) came up with a brightness ratio model that looked like this:
SEP Latitude < Year > TB/<TB> [K] rms [%]
-85.0 1986.70 1.05 5.0
-70.0 1991.92 1.06 3.6
-55.0 1994.16 1.03 6.7
-40.0 1998.17 1.00 6.5
-25.0 2001.67 0.99 5.5
-10.0 2004.55 0.96 5.6
  • Another paper, Weiland et al. (2010, http://fr.arxiv.org/abs/1001.4731) used the WMAP 7-year data to calibrate the major planets, including Uranus and Neptune, using the CMB.
    • they observed near all the ATCA bands
      • K: 23 GHz
      • Ka: 33 GHz
      • Q: 41 GHz
      • V: 61 GHz
      • W: 94 GHz
    • Over 7 years (2001-2008), the sub-WMAP latitude changed only from -26.4 degrees to 4.4 degrees, and the data quality was such that no trend with sub-WMAP latitude could be determined
    • The mean absolute brightness temperature of Uranus over these 7 years is tabulated below:
Frequency (GHz) Mean TB [K] Uncertainty [K]
23 168 11
33 128 9
41 155 6
61 140 5
94 121 4
  • Note that there is a clear "dip" in temperature at 33 GHz, which we will almost certainly have to compensate for with our Uranus model
    • believed to be due to atmospheric phenomena on Uranus

we need to determine whether atmospheric effects create a large uncertainty

we need to determine whether Uranus' internal temperature changes enough to require periodic calibration

do we need to make a complex model of Uranus so that different contributions of polar and equatorial regions can be accounted for on different baseline lengths?

Procedure options

  1. Modelling - try to find or develop a model that correctly accounts for the orientation of Uranus, and implement this in MIRIAD
  2. Absolute calibration - periodically measure Uranus' brightness temperature directly, and use that when calibrating data in MIRIAD
  3. Relative calibration - use another source with known flux (eg. Neptune) that may not be as directly usable but is more stable, to calibrate Uranus periodically

Procedures

There seems to be enough information to make a decent model of Uranus' brightness temperature over the entire ATCA mm frequency range.

To start, we can compare the values given by Weiland et al. (2010) to those plplt reports for the same frequencies:

Frequency (GHz) Band Name Weiland TB [K] plplt TB [K] Ratio plplt/Weiland
23 K 168 +- 11 174 1.04
33 Ka 128 +- 9 162 1.27
41 Q 153 +- 6 155 1.01
61 V 138 +- 5 145 1.05
94 W 120 +- 4 132 1.1

The problem with this model is that for the calculation of the mean, Weiland et al. (2010) does not take into account the sub-WMAP latitude.

We want a model of how Uranus' brightness changes with sub-Earth (or sub-whatever latitude), but I think Kramer got the latitude values wrong. This is because for 1986.7 and nearby, the sub-Earth latitude was nowhere near -85 degrees, let alone straddling it. So let's do it again with the semi-raw data:

Year TB/<TB> rms [%] N Start MJD (01-01) End MJD (31-12) SEL (01-01) SEL (31-12) Mean SEL Min SEL Max SEL
1986 1.05 4.6 8 46431 46795 -81.953 -80.092 -81.482 -82.126 -80.092
1987 1.04 6.1 5 46796 47160 -80.035 -77.184 -79.155 -80.542 -77.184
1991 1.08 2.7 6 48257 48621 -66.439 -62.760 -64.486 -66.439 -62.574
1992 1.06 2.6 12 48622 48987 -62.703 -58.941 -60.522 -62.703 -58.549
1993 1.06 6.7 14 48988 49352 -58.884 -55.166 -56.531 -58.827 -54.514
1994 0.99 3.9 10 49353 49717 -55.109 -51.385 -52.547 -55.053 -50.477
1995 0.98 7.1 11 49718 50082 -51.329 -47.603 -48.563 -51.273 -46.445
1996 0.99 6.6 14 50083 50448 -47.548 -43.769 -44.592 -47.548 -42.422
1997 0.99 7.3 16 50449 50813 -43.715 -39.994 -40.617 -43.715 -38.409
1998 1.00 6.4 18 50814 51178 -39.942 -36.224 -36.653 -39.889 -34.409
1999 1.03 5.0 19 51179 51543 -36.173 -32.460 -32.711 -36.122 -30.424
2000 1.03 7.4 21 51544 51909 -32.411 -28.654 -28.784 -32.361 -26.453
2001 1.00 3.7 38 51910 52274 -28.606 -24.902 -24.872 -28.606 -22.498
2002 0.97 4.0 32 52275 52639 -24.856 -21.151 -20.973 -24.856 -18.556
2003 0.97 4.5 19 52640 53004 -21.108 -17.399 -17.084 -21.108 -14.626
2004 0.97 4.7 28 53005 53370 -17.359 -13.606 -13.206 -17.359 -10.704
2005 0.95 7.2 14 53371 53735 -13.567 -9.846 -9.320 -13.567 -6.788

We can get the sub-Earth latitudes as a function of MJD from http://pds-rings.seti.org/tools/ephem2_ura.html, and the full daily ephemeris for 1986-01-01 to 2005-12-31 that was used to obtain these figures is attached to this page.

Now we can make a fit of mean sub-Earth latitude vs flux ratio from this data. Because we can think of the planet as two regions - the polar region and the equatorial region - we could justify going to a two-component fit. However, it is likely that the temperature gradient between the two regions is fairly uniform, so it may not need to be that complicated. Instead, we use a second-order fit to the data. We weight each year's point by the rms percentage value. The plot of the fit can be seen below:

kramer_90ghz_subearthlat_fit.png

The equation of the fit is:

(1)
\begin{align} ratio = 0.929029 + 0.00275615 \times \sqrt{(l_{se}^{2})} - 1.28851\times 10^{-5} \times l_{se}^{2} \end{align}

where $ratio$ is the ratio between the observed temperature and the expected temperature (what Kramer et al. (2008) calls TB/<TB>), and $l_{se}$ is the sub-Earth latitude in degrees.

We can see that it depends only on the absolute value of the sub-Earth latitude, since there is (currently) no reason to believe that the two polar regions will have a different temperature. This assumption needs to be made however because we have no data that views the northern polar regions - all this data comes from looking at the southern pole.

To test this fit, we see how it performs compared to the Weiland et al. (2010) data. Below we plot their seasonal data, and split the plot by band. In each plot, the fit is:

(2)
\begin{align} T_{b} (l_{se}) = <T_{b}> \times ratio \end{align}

where $T_{b}$ is the expected brightness temperature at some $l_{se}$ (or in this case the sub-WMAP latitude $D_{W}$), $<T_{b}>$ is the mean brightness temperature for that band as determined by Weiland et al. (2010), and $ratio$ is calculated using the equation above.

weiland_wmapbands_fitoverlay.png

This is a very good fit, and shows that Weiland et al. (2010) was right when they considered the $D_{W}$ range to be too small to make a fit - the fit line is basically flat through this region.

So, it looks like we have a model that adequately predicts a brightness temperature scaling factor given the Uranus viewing angle!

The next step is to try and make a model that interpolates between the frequency bands that Weiland et al. (2010) has observations for; ie. we need a model for $<T_{b}>$.

Below is a plot of the mean brightness temperature determined by Weiland et al. (2010) as a function of frequency. The green line is a fit that uses all points except the one at 33 GHz, which is clearly and significantly below the line of best fit. The question is then, do we see this in our ATCA data? Is this the result of a very narrow dip in the spectrum, or is it a very gradual dip? We need to know this information in order to correctly estimate Uranus' brightness temperature in this region.

uranus_powerspectrum_fit1.png

To determine the location and significance of this dip, I devised an observing program that looked at Uranus and Neptune on the same day at similar elevations. This program was carried out on Saturday 27 March 2010, and used 1921-293 as a bandpass calibrator, 0003-066 as a phase calibrator for Uranus and 2155-152 as a phase calibrator for Neptune. The observing band centres were 21/23 GHz, 31/33 GHz, 35/37 GHz, 39/40 GHz, 41/43 GHz and 45/47 GHz.

Reduction was complicated by a finding that there was no way to correct for spectral index across the 2 GHz band with MIRIAD. Once a bandpass calibration had been made with mfcal, mfboot/gpboot would only scale the flux by a single number and would not take into account the spectral slope of the flux calibrator. Mark W. altered mfboot to correct this deficiency, and this reduction is made with the new mfboot.

Reduction was:

  • mfcal 1921 to get bandpass calibration
  • gpcopy 1921 to 0003 and 2155 for bandpass calibration
  • mfcal 0003 and 2155 with "options=nopassol" to get phase calibration
  • gpboot 0003 and 2155 back to 1921's flux calibration
  • gpcopy 0003 to Uranus, 2155 to Neptune for bandpass and phase calibration
  • mfboot Neptune, Uranus, 1921, 0003 and 2155 to Neptune's flux calibration (as we assume Neptune has no spectral dip)

Using uvspec to get the spectral data, I plotted the spectra of each source over the full 20-48 GHz range to check for consistency. This plot is shown below.

neptune_scaling.png

From this we can see that there is no significantly visible dip in Uranus' spectrum anywhere in the frequency range we observed. We can be confident about our calibration since the spectral behaviour of 1921, 0003 and 2155 are all realistic and believable, with no jumps or bumps or horrible discontinuities. If there is a dip in Uranus flux, then either all the sources here share that dip or it is not significant enough to show up on this plot, and therefore probably not important for calibration purposes.

Time to implement the new Uranus model into MIRIAD. The routines plpar and plphyeph together give the information necessary to get a fairly accurate measure of Uranus' sub-Earth latitude (which I have found to be within a degree of the currently most accurate value). This information is used to get the flux multiplication factor by the fit I made above, and the brightness temperature itself is determined using the straight line fit:

(3)
\begin{align} log (T_{b}) = 2.58533 - 0.253313 log (\nu) \end{align}

where $\nu$ is the frequency in GHz. This fit was made with the Weiland et al. (2010) data excluding the 33 GHz point, and is simply a linear least-squares first order fit.

old stuff
We'll start with procedure 2, as if it works, it has a good chance of allowing us to use current data to estimate how Uranus' brightness has been changing, allowing for correct recalibration, and may also allow us to develop a good model that may be applicable in the future.

My first idea is this:

  • Take observations of Uranus at 3mm with the ATCA
    • Because 3mm ATCA observations are accompanied by paddle scans, which set the absolute brightness temperature scale of the receiver, we can essentially make a direct measurement of Uranus' brightness temperature at that time.
  • Using models of how Uranus' brightness temperature varies with frequency, it may be possible to predict from these 3mm data the temperatures at other frequencies
    • To test this, I will try to calibrate 1934-638 at 12mm with the temperature derived from Uranus models and the 3mm data. If the new calibration matches the old calibration, it may be a good indication that this procedure might work.

For this test, I need data taken in close temporal proximity to each other for Uranus at 3mm, Uranus at 12mm and 1934-638 at 12mm. Thankfully, using the new observation database, I have found several datasets that may be suitable.

Set 1:

  • 3mm Uranus data on 2009-06-25, project C007, H75 array
  • 12mm Uranus data on 2009-06-26, project C1730, H75 array
  • 12mm 1934-638 data on 2009-06-26, project C1730, H75 array

Set 2:

  • 3mm Uranus data on 2009-09-(11-12), project C1800, H214 array
  • 12mm Uranus data on 2009-09-10, project C007, H214 array
  • 12mm 1934-638 data on 2009-09-10, project C007, H214 array

Set 3:

  • 3mm Uranus data on 2009-09-15, project C2102, H214 array
  • 12mm Uranus data on 2009-09-15, project C2046, H214 array
  • 12mm 1934-638 data on 2009-09-15, project C2046, H214 array

The procedure goes like this:

  • Check that there is a paddle directly before Uranus: paddletimes.pl blah.C007, discard if not suitable
  • Load all the data: atlod in=blah.C007 out=c007_blah.uv options=birdie,rfiflag,xycorr,opcorr,noauto
  • Apply fixes: uvfix vis=c007_blah.uv out=c007_blah.fixed.uv tsyscal=any
  • Split into source files: uvsplit vis=c007_blah.fixed.uv
  • Beginning with the 3mm Uranus data:
    • flag any bad data in the 3mm bandpass calibrator
    • split the appropriate 3mm bandpass calibrator into 128 MHz chunks using uvsplit: uvsplit vis=1921-293.93000 maxwidth=0.128
    • split Uranus into 128 MHz chunks as well: uvsplit vis=uranus.93000 maxwidth=0.128
    • calibrate the bandpass calibrator: dac_repeat.pl —task mfcal —repeat vis —repeat-freq 1921-293 interval=0.1 refant=2
    • after checking for validity, copy bandpass to Uranus (excluding gains calibration, as we want to keep the paddle calibration): dac_repeat.pl —task gpcopy —repeat vis —repeat-freq 1921-293 —repeat out —repeat-freq uranus options=nocal
    • use mfboot to get the scaling factors for each frequency chunk: dac_repeat.pl —task mfboot —repeat vis —repeat-freq uranus —repeat device —repeat-freq uranus_mfboot_%f.ps/cps options=noapply
    • make note of all the scaling factors, and whether they appear valid by looking at the fit in the uranus_mfboot_*.ps files

Results

Set 1

File was 2009-06-25_1436.C007. Uranus observed at 17:10:44, bandpass calibrator was 1921-293, appropriate paddle time was 16:42:44 (for 1921-293). Since almost half an hour passed between paddle and Uranus observation, I'm not certain this dataset will be useful.

Frequency (MHz) Uranus 3mm scale factor plplt Uranus Tb (K)
92040 1.379 133
92168 1.195 133
92296 1.272 133
92424 1.215 133
92552 1.119 133
92680 1.157 133
92808 0.994 133
92936 0.946 133
93064 1.101 133
93192 1.402 133
93320 1.655 133
92448 1.790 132
93576 2.260 132
93704 2.428 132
93832 2.673 132
93960 3.003 132

Set 2

These are spread over two days, 2009-09-11 and 2009-09-12.

On 2009-09-11:

Files are 2009-09-11_1554.C1800 & 2009-09-11_1619.C1800 Uranus observed at 16:31:04, bandpass calibrator was 2223-052, appropriate paddle time was 16:28:54 (on Uranus).

On 2009-09-12:

Files are 2009-09-12_1532.C1800 & 2009-09-12_1639.C1800 Uranus observed at 16:54:34, bandpass calibrator was 2223-052, appropriate paddle time was 16:52:24 (on Uranus).

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