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Geometry/Planetary Information
![](http://www.humboldt.edu/~math/currentevents/colloquim/archives/Spring,'02/image012.gif) |
Constant Values Used [The following value of
each constant was used throughout this project. These values are from my
notes.]
|
Newton's Gravitational Constant (G) |
6.6720E-11 |
Boltzman's Constant (k) |
-2.2860E+02 dBW/K/Hz or 1.3800E-23 J/K |
Speed of Light (c) |
3E+08 m/s |
Mass: |
1.0244E+26 kg |
Radius: |
2.4764E+04 km |
Sidereal Day: |
5.7996E+04 seconds |
Max Distance to Earth: |
4.69E+12 m |
![](D_Geometry_files/image002.gif) |
Attenuation Next, I solved for the maximum
attenuation that could be experienced. This was accomplished by solving
for the equatorial probe first. Then I performed some geometric
calculations to find the maximum distance through which the +/- 45
degree probes must transmit and adjusted my calculations accordingly. As
was done throughout this project, a philosophy was adopted to solve for
the worst case scenario, knowing that the design will work under all
other conditions, while maximizing robustness. The probes begin at
Neptune's gaseous surface and proceed down to a depth of 50 bar (which
was experimentally determined to be 420 feet). The raw data was provided
as an Excel spreadsheet and was compiled by Georgia Tech researcher
Priscilla Mohammed. The expanded Excel file with calculated attenuation
curves is included as NeptuneAbsorption.xls |
The data provided was given in 2 km slices, which then had to be accumulated
for the primary gases that comprise Neptune's atmosphere. These gases are
Hydrogen, Helium, Hydrogen Sulfide, Ammonia, Water, Methane, and Phosphene.
Attenuation information was only provided for Water, Ammonia, Hydrogen Sulfide,
Phosphene, and Hydrogen. Furthermore, these calculations were performed for 100
MHz, 500 MHz, 1 GHz, and 2.3 GHz.
Using geometry and trigonometry, I was able to find the maximum distance
transmitted through by the
+/- 45 degree probes . These calculations can be found in the
included Excel file NepProjCalc.xls.
![](D_Geom2.gif)
![](D_Geom3.gif)
The following is the resultant attenuation curve for the 45 degree probes.
![](D_Geom1.gif)
Again viewing the design from a worst-case scenario, I limited
my view of the curve to the 420 m case (far right). I wanted to minimize the
attenuation I experienced, also keeping in mind that there are no known sources
of interference around Neptune. Based upon this logic and a cursory look into communication
components, I decided to use a probe transmit frequency of 300 MHz. I then
interpolated by using 1/4 the 100 MHz value and 3/4 the 500 MHz value (as the
plot is not linear) to find the attenuation at this frequency. Thus, at 50 bar
(420 km) below Neptune's surface, the maximum attenuation experienced in a 300
MHz communication link is 3.66 dB.
![](D_Geometry_files/image002.gif)
![](D_Geom1.jpg) |
Calculating Geostationary Neptunian Orbit
Kepler’s third law of planetary motion provides us with the necessary
formula for calculating the period of an orbiting body (Pratt 23):
T^2=(4π^2a^3)/μ
NASA lists one Neptune sidereal day as 16.11 hours, or 57,996
seconds, which we will use as the period, or T.
Mu (μ) is equal to Neptune’s mass (1.0244 x 1026 kg) times Newton’s
gravitational constant (6.672 x 10-11 Nm2/kg2), or 6.8347968 x 1015.
Thus, Neptune’s geostationary orbit radius is 83,506.6 (83,506.5710487)
km. |
Geometry - Probes -
Relay Satellite - DSN
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