Dewey, I'd also like to thank you for showing me that there's (almost) always time for ice cream even when there's no time for ice cream, and that you can't convince me to give up radio astronomy to sell shoes (albeit only history will tell if you should have tried harder). These dielectric constant calculations lead to estimates of subsurface density as a function of latitude.
INTRODUCTION
The size of the North Polar Cold Region was found to be consistent with measurements of the extent of the North Polar Cap made visually during the same season in previous years by other observers (e.g. Iwasaki et al and James. In addition to the whole disc measurements and interpretation of the polar region, we also used radio maps of the data, made using standard NRAO software, to fit a thermo-radiative model.
DATA ACQUISITION AND CALIBRATION
Mapping and Self-Calibration
The northern data were calibrated using the 3C286 compact radio source as the primary calibrator with an assumed flux density of 7.41 Jansky at 6.14 cm and 3.45 Jansky at 2.00 cm.
MARS: 6CM SOUTH DATA SET
THE MODELS
- Whole-Disk Models
- Polar Cold Region Models
- Thermal and Radiative Models
This is because the brightness temperature of the entire disk is the value of the visibility function at zero spacing, that is, the intersection with the side axis. J0(A) is a constant that can be estimated from the brightness temperature fit of the entire disk given above.
NORTH DATA SET: RESULTS AND DISCUSSION
Whole-Disk Results
Much of the discrepancy present can be attributed to the fact that different observations were made in different seasons, at different times under the Earth and at different wavelengths. The accuracy with which the brightness temperatures of the whole disk in the model (estimated from the thermal modeling using the fitted dielectric constants of the whole disk) match the actual brightness temperatures of the whole disk will play a crucial role.
North Polar Cold Region
This means that sometime during the year the surface becomes much warmer than the C02 sublimation temperature (Keiffer et al., 1977). Assuming that C02 transport is relatively efficient, the surface temperature remains close to the C02 sublimation temperature. Solutions to the heat equation under these circumstances show that, at a latitude of about 65°N, the temperature at a depth of a seasonal thermal skin depth, about llOcm, will be about 10 to 15 degrees warmer than the temperature of C02 sublimation during the season in which our measurements were made.
The solid line corresponds to the season during which the North data set was taken.
PHYSICAL TEMPERATURE (K)
Thus, it is not surprising that the Cold North Region temperatures are warmer than the sublimation temperature of C02, and that the 6 cm results are warmer than the 2 cm results. Because the thickness of the seasonal C02 cap is expected to increase with latitude, you would expect the C02 freezing layer to contribute increasingly to the brightness temperature as latitude increased. Moreover, at higher latitudes, the seasonal cap is present for more of the year, and the subsurface temperature gradient is, therefore, smaller.
All these factors together imply that the radio brightness temperature at the higher latitudes will be almost that of the C02 frost alone.
I) 3'HnlvHadial ssaNlHDIHH
Latitudinally Binned Results
The two extremes of the range of models fitted to the data are also shown. The strong curvature at southern latitudes is due to the effects of the beam shape function and the radio emissivity. The slope of the smooth edge of the model agrees reasonably well with the data in the 2 cm bin.
This implies that, as seen at 2 cm, the edge of the north polar cold region is a sharp boundary.
The results here are similar to the radio absorption wavelengths found for the subsurface of the Moon. Both thermal inertia and dielectric constant depend on the density of the substrate. The effect of varying the radioabsorption length in the model is to sample a different region below the surface.
The outer error bars are an additional, combined error due to the dispersion in the dielectric constant of the powders of rocks of different types.
SOUTH DATA SET: RESULTS AND DISCU SSION
Whole-D isk Results
Recall that although the brightness temperatures of the entire disk were obtained from the recalibrated data, the dielectric constants of the entire disk were not. This is because the polarized flux is divided by the unpolarized flux to obtain the dielectric constant of the entire disk. Comparing the effective dielectric constants over the entire disk from the South data set with those derived from the North data set shows that the dielectric constant at each wavelength is about 0.3 less than that of its counterpart in the North.
The lower effective dielectric constants of the entire disk derived from the southern data set are consistent with the north–south asymmetry seen in the thermal inertia and albedo maps of Palluconi and Keiffer (1981) .
South Polar Cold Region
Recall the very simple model used to fit the temperature and illumination extent of the Cold South Polar Region. This region of lower brightness temperature is the reason that the cold south polar region is much larger than the visual extent of the South Polar Cap, as determined by other observers. The South Polar Cap is at a different distance from the geoid than implicitly assumed in the thermal model.
It may be possible that the same physical causes may explain the observed extent of the South Polar Cold Region and the fact that there is a residual C02 frost cover.
Latitudinally Binned Results
The first example consisted of a 4 cm region with a thermal inertia of 6.5 (the planetary average used by Keiffer et al., 1977) overlying a region with a thermal inertia of 4.0. The dashed line is the same calculation, but with a thermal inertia of 10.0 below 4 cm of a subsurface with a thermal inertia of 6.5. The dashed line is the same calculation, but with a thermal inertia of 10.0 below 4 cm of a subsurface with a thermal inertia of 6.5. <.
The dotted line is the same calculation, but with a thermal inertia of 10.0 based on 4cm of subsurface with a thermal inertia of 6.5. little effect on surface temperatures.
I) 3:HilJVH3dW3J, SS3NJ,HDIH8
A latitudinal average of the radio absorption longitude would give an average of about 15, the same as for the northern data set. North of the equator, the estimated wavelengths of radio wave absorption clustered exactly around 15-X, which is roughly what was determined for the northern data set. A comparison of the dielectric constants in the region between 30°N and 20°S shows that at both wavelengths the dielectric constants derived from the northern data set are larger than the corresponding dielectric constants derived from the southern data set.
As with the North data set, the dielectric constants can be used to obtain an estimate of the effective subsurface density.
WHOLE-DISK BRIGHTNESS STUDIES
Variation with Dielectric Constant
The models were run with four different dielectric constants for each of the ten different radioabsorption wavelengths. It is relatively easy to see from Figure 6.4 that for the selected wavelength all the curves are similar. Other curves not included show the same similarity for each of the radioabsorption wavelengths investigated.
The numbers in Table 6.2 are calculated by dividing the curve for t = t2 = 2.2 by each of the remaining curves.
Variation with Longitude
The seasonal curves were all divided by the curve at 75°W to obtain ratios that can be used to easily get from the nominal model to a whole-disk brightness temperature at a different radio absorption longitude and different subsurface longitude. This is why the nominal model has a subsurface longitude of 75°W, this longitude has a seasonal brightness temperature curve which is near the median of the seasonal brightness temperature curves for all longitudes. As with the variations due to changes in the dielectric constant, the variations due to changes in subsurface longitude are subject to the caveat that the thermal parameters above 60°N and below 60°8 are estimated from the adjacent latitudes because the data of Palluconi and Keiffer ( 1981) do not extend into these regions.
Therefore, these regions will have smaller weights when averaged to obtain the entire disc brightness temperature and should not cause problems for low subsurface latitudes.
Variation with Time of Day
For a wavelength of 6 cm (radio absorption length of 90 cm) the variation is just over 1 degree over this range of subterranean time of day. This means that the assumption that the brightness of the entire disk does not change as the phase angle changes implies an error of about 2 percent at 2 cm.
Variation with Sub-earth Latitude
The crosses are the result for a subterranean latitude of 25°N and the x's are for a subterranean latitude of 25°S. Tables 6.5a-f (at the end of this chapter) summarize the temperature variations as a function of season and radio absorption length for seven different subterranean latitudes. The albedo of the rock surface and the thermal inertia of the surface and, by extension, the subsurface, were determined by averaging over all longitudes the values for these parameters for the latitude adjacent to these limits.
Therefore, of all the effects mentioned in this chapter, the subsurface latitude is not only the most important geometric parameter, it is also the one with the largest amount of error inherent in the model.
Discussion
From the measurements presented in Chapters 4 and 5, a good estimate of the radio absorption length is 30 cm at a wavelength of 2 cm and 90 cm at a wavelength of 6 cm. Since this estimate is made assuming that the radio absorption length is scaled as wavelength, a good estimate of the radio absorption length at any wavelength is 15>. A realistic estimate of the error in the dielectric constant changes the brightness temperature of the entire disc by a significant amount.
In addition, variations in dielectric constant over the planet's surface are large enough to contribute to brightness temperature variations with different geometries, as do variations in thermal inertia.
IMPLICATIONS FOR FUTURE RESEARCH
BIBLIOGRAPHY
The Longitudinal Variation of the Thermal Inertia and of the 2.8 Centimeter Brightness Temperature of Mars Astrophysical Journal. In Image Formation from Coherence Functions in Astronomy, Proceedings of IAU Colloquium No.49 (C. Van Schooneveld Ed.) pp.261-275, D.
APPENDIX A
THERMAL MODEL DETAILS
This is because one of the main tests of the thermal model was its ability to replicate the work of Keiffer et al. This surface temperature gradient is a derivative of a Taylor expansion of the tern perature at the first depth level. That is, instead of the temperatures at n + 1, the temperatures at n are used to estimate the surface thermal gradient, i.e.
The surface temperature plots are identical to those whose main time step was five days.
