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CHAPTER 5. SPIRE FLUX CALIBRATION
In calculating the planetary angular sizes and solid angles, a correction is applied for the
inclination of the planet’s axis at the time of observation, and the apparent polar radius is
given by (Marth, 1897):
⇥
⇤1/2
rp a = req 1 e2 cos2 ( )
,
(5.1)
where
is the latitude of the sub-Herschel point, and e is the planet’s eccentricity:
"
2
req
rp2
e=
2
req
#1/2
.
(5.2)
The observed planetary disc is taken to have a geometric mean radius, rgm , given by
rgm = (req .rp
a)
1/2
.
(5.3)
For a Herschel-planet distance of DH , the observed angular radius, ✓p , and solid angle, ⌦, are
thus
rgm
✓p =
and ⌦ = ⇡✓p2 .
(5.4)
DH
Typical angular radii for Uranus and Neptune are 1.700 and 1.100 respectively.
5.1.2
Neptune and Uranus models
Models of Uranus and Neptune have been agreed by the Herschel Calibration Steering Group
(HCalSG) as the current standards for Herschel, and are available on the HSC calibration
ftp site1 . The models currently used for SPIRE are the “ESA-4” tabulations for both Uranus
(based on Orton et al. 1986, 2014) and for Neptune (based on the updated model of Moreno
1998, 2010). The absolute systematic flux uncertainty for Neptune is estimated to 4% (R.
Moreno, private communication), while comparing Uranus and Neptune, the absolute uncertainties are of the order of 3% (Swinyard et al., 2014).
The Uranus and Neptune disk-averaged brightness temperature spectra are plotted in Figure 5.1. In the Herschel range, the disk averaged brightness temperatures increase with
wavelength as deeper layers in the atmosphere are probed. The planets have similar temperatures, despite Neptune’s greater distance from the Sun, because Neptune has an internal heat
source. This also leads to a more dynamic neptunian atmosphere resulting in some prominent
spectral features whereas the Uranian spectrum is largely featureless (see e.g. Teanby & Irwin
2013).
Typical photometer (250, 350, 500) µm calibration flux densities (see Section 5.2 for precise
definition) are (160, 100, 60) Jy for Neptune and (370, 250, 150) Jy for Uranus.
5.1.3
Mars models
Web-based models of the martian continuum by Emmanuel Lellouch and Bryan Butler are
available at http://www.lesia.obspm.fr/perso/emmanuel-lellouch/mars/ and http://
www.aoc.nrao.edu/~bbutler/work/mars/model/.
1
ftp://ftp.sciops.esa.int/pub/hsc-calibration/PlanetaryModels/