SOLUTION TO THE VOYAGER II TRANSMITTER PROBLEM

Calculate the strength of Voyager II's X-Band telemetry signal when received on earth from the vicinity of Jupiter, Saturn, Uranus, and Neptune. Assume a transmitter output power of 21.3 watts. Express your answers in watts per square meter. Show that the calculated power outputs confirm signal weakening by the inverse square law relationship?

POWER FLUX DENSITY (PFD) = P x G / 4 x pi x r²

PFD = watts per meter²

P = 21.3 watts (X-Band transmitter output)

G = 6.5 x 10⁴ (antenna gain)

PG = ERP (effective radiated power)

r = radius or distance from earth

APPROXIMATE DISTANCE BETWEEN EARTH AND JUPITER

5.2 A.U. – 1 A.U. = 4.2 A.U. x 93,000,000 = 391 million miles

391 million miles x 1609.344 meters per mile = 6.3 x 10¹¹  meters = r1

APPROXIMATE DISTANCE BETWEEN EARTH AND SATURN

9.6 A.U. – 1 A.U. = 8.6 A.U. x 93,000,000 = 800 million miles

800 million miles x 1609.344 meters per mile = 1.3 x 10¹² meters = r2

APPROXIMATE DISTANCE BETWEEN EARTH AND URANUS

19.2 A.U. – 1 A.U. = 18.2 A.U x 93,000,000 = 1.7 billion miles

1.7 billion miles x 1609.344 meters per mile = 2.7 x 10¹² meters = r3

APPROXIMATE DISTANCE BETWEEN EARTH AND NEPTUNE

30.1 A.U. – 1 A.U. = 29.1 A.U. x 93,000,000 = 2.7 billion miles

2.7 billion miles x 1609.344 meters per mile = 4.3 x 10¹² meters = r4

POWER GAIN OF VOYAGER II'S X-BAND TRANSMITTER AND ANTENNA

21.3 watts x 6.5 x 10⁴ antenna gain = 1,384,500 watts (effective radiated power)

VALUE OF 4 x pi

12.5664 = k1

POWER OF VOYAGER II'S X-BAND TRANSMISSION FROM JUPITER

PG / k1 x (r1)² = 1,384,500 / (12.5664) x (6.3 x 10¹¹  meters)²

1,384,500 / (7.9 x 10¹²)² = 1,384,500 / 5 x 10²⁴

2.8 x 10^-19  watts/meter²

POWER OF VOYAGER II'S X-BAND TRANSMISSION FROM SATURN

PG / k1 x (r2)² = 1,384,500 / (12.5664) x (1.3 x 10¹² meters)²

1,384,500 / (1.6 x 10¹³)² = 1,384,500 / 2.1 x 10²⁵

6.6 x 10^-20 watts/meter²

POWER OF VOYAGER II'S X-BAND TRANSMISSION FROM URANUS

PG / k1 x (r3)² = 1,384,500 / (12.5664) x (2.7 x 10¹² meters)²

1,384,500 / (3.4 x 10¹³)² = 1,384,500 / 9.2 x 10²⁵

1.5 x  10^-20 watts/meter²

POWER OF VOYAGER II'S X-BAND TRANSMISSION FROM NEPTUNE

PG / k1 x (r4)² = 1,384,500 / (12.5664) x (4.3 x 10¹² meters)²

1,384,500 / (5.4 x 10¹³)² = 1,384,500 / 2.3 x 10²⁶

6 x 10^-21 watts/meter²

RATIO OF JUPITER TO SATURN DISTANCE

800 million miles / 391 million miles = 2.05:1

(2.05)² = 4.2

The signal from Jupiter should be 4.2 times stronger than the signal from Saturn

2.8 x 10^-19  watts/meter² / 6.6 x 10^-20 watts/meter² = 4.2

RATIO OF JUPITER TO URANUS DISTANCE

1.7 billion miles / 391 million miles = 4.35:1

(4.35)² = 18.9

The signal from Jupiter should be 18.9 times stronger than the signal from Uranus

2.8 x 10^-19  watts/meter² / 1.5 x  10^-20 watts/meter² = 18.7

A 1% error due to rounding values

RATIO OF JUPITER TO NEPTUNE DISTANCE

2.7 billion miles / 391 million miles = 6.9: 1

(6.9)² = 47.7

The signal from Jupiter should be 47.7 times stronger than the signal from Neptune

2.8 x 10^-19  watts/meter² / 6 x 10^-21 watts/meter² = 46.7

A 2% error due to rounding values

COMMENT:

Voyager II's X-Band transmitter signals, as received on earth, were extraordinarily weak, although by using very large parabolic antennas scattered across the world and networked together, they could not only be received successfully but decoded into spectacular photographs from the far end of the solar system. The success of this system represented a minor triumph of radio telemetric technology. The X-Band antenna had to be pointed precisely at earth. Even a misalignment of as much as 0.02% at the distance of Neptune would have been enough to decrease signal strength by as much as half. The ratio calculations above clearly show signal strength decreasing by the inverse square relationship as distance from earth increases, confirming the general accuracy of the power calculations.