"Penetrating Planets Since 2004"
ECE6390 Introduction to Satellite Communications Fall, 2004
 Initial Research
 Introduction
 Project Requirements
 Neptune Facts
 Calculations
 Penetrating Probes
 Introduction
 Design
 Calculations
 Parts & Pricing
 Relay Satellite
 Introduction
 Link with Probe
    Calculations
 Link with Earth
    Calculations
 Parts & Pricing
 Conclusions
 Summary
 Total Cost
 Contact Info
 Home » Initial Research » Calculations

Calculations

 

The relay satellite in Geostationary Neptune Orbit (GNO) is the heart of the communications link between the probes and the Earth.  The satellite will act as a data gatherer and will transmit this data to the Earth whenever the satellite is in view of the Earth.  The satellite has been decided to be placed in equatorial orbit; the distance required for GNO is determined with the mass and period of Neptune:


FormBox[RowBox[{T, =, RowBox[{(2 π R^(3/2))/μ^(1/2) R, =, RowBox[{((Tμ^(1/2))/(2π))^2/3, =, RowBox[{8.35039, , 10^7, m}]}]}]}], TraditionalForm]


The radius of GNO was calculated to be 83,503.9 km,  approximately three times the equatorial radius.  As a result of this large distance, the satellite will have to travel at 9.04706km/sin order to maintain its orbit and not be sucked in by Neptune's gravity.  Figure 1 shows a diagram of Neptune with a geostationary orbit.


[Graphics:/mathematica/HTMLFiles/index_6.gif]

Figure 1: Geostationary Neptune Orbit             


While the distance of free space between the satellite and the 0° probe can be calculated simply by subtracting the GNO radius and the equatorial radius, calculating the distance to the off-angle probes requires the use of the law of cosines


R_probe = (R_equatorial^2 + R_GNO^2 - 2R_equatorialR_GNOCos(45))^(1/2)  FormBox[RowBo ... bsp;     , RowBox[{= 68, ,, RowBox[{676.7,  , km}]}]}], TraditionalForm]


The angle from the relay satellite to the off-angle probe can be determined by


FormBox[RowBox[{Angle, =, RowBox[{cos^(-1)((R_probe^2 - R_equatorial^2 + R_GNO^2)/(2 R_probe R_GNO)), =, RowBox[{14.8606, }]}]}], TraditionalForm]


This angle is important so that we can choose a dish antenna whose Half-Power Beamwidth (HPBW) will be close to approximately 30°.

 

The bit error rate (BER) determines the probability that a bit has incorrectly been received at the receiver that defines the quality of a digital communications system, much like the carrier to noise ratio in an analog system [1].  The statistical probability that a bit is incorrect often uses the Q-function, which is a "normalized form of the cumulative normal distribution function giving the probability that a variate assumes a value in the range [0, x]" [2].  The Q-function is defined by


Q(x) =1/(2π)^(1/2) ∫_x^∞^(-t^2/2) t


For an uncoded bi-polar phase shift keying (BPSK) signal, the BER is determined by


BER_BPSK = Q((2C/N)^(1/2))


where C/N is the carrier to noise ratio at the receiver in linear form.  Using a turbo code method of forward error correction essentially allows for an error-free communications link if the received C/N is greater than about 1 or 2 dB (see Penetrating Probes » Design).

 

 

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References:
1. Pratt, Bostian, Alnutt. Satellite Communications, Second Edition. Copyright 2003.
2. Wolfram MathWorld (http://mathworld.wolfram.com/NormalDistributionFunction.html)

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William W
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