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Ranging to Mars

Space and time · level 6, lesson 6 of 6

A question to start

How can a clock on Earth measure the distance to a spacecraft on Mars?

The idea

Send a radio signal to the spacecraft. It sends one straight back. Time the round trip \(t\), and

\[d = \frac{c\,t}{2},\]

with \(c\) the speed of light, 299,792.458 km/s. This is called ranging. It is the laser trick from the Moon, done with radio.

Because \(c\) is so large, a small timing error makes a big distance error. One microsecond, a millionth of a second, of round-trip time is \(299{,}792 \times 0.000001 \div 2 \approx 0.15\) km, or 150 m.

The Viking spacecraft, which reached Mars in 1976, returned radio signals sent from Earth. Ranging to them was so precise that, near the 1976 conjunction, scientists could see the signals arrive later than plain distance allows: delayed by up to about 250 microseconds, as Einstein's theory of gravity predicts. That story is in level 8.

signal outsignal backdistance = c × round-trip time ÷ 2

Worked examples

1. A round trip of 20 minutes. \(t = 1{,}200\) s. \(d = 299{,}792 \times 1{,}200 \div 2 \approx 180\) million km.

2. Timing to a nanosecond. A billionth of a second of round-trip time is \(299{,}792{,}458 \times 10^{-9} \div 2 \approx 0.15\) m: 15 cm.

3. The Viking delay as a distance. 250 microseconds of extra round-trip time looks like \(299{,}792 \times 0.000250 \div 2 \approx 37\) km of extra distance, far too much to be a mistake.

Try it

  1. Why divide by 2 in the ranging formula?
  2. A round trip takes 10 minutes. How far away is the spacecraft?
  3. Round-trip timing is good to 10 nanoseconds. About how precise is the distance?
  4. What did Viking ranging show near conjunction?
Answers
  1. The time is for there and back; the distance is one way.
  2. \(299{,}792 \times 600 \div 2 \approx 90\) million km.
  3. \(10 \times 0.15 = 1.5\) m.
  4. Signals passing near the Sun arrived late, by up to about 250 microseconds, as general relativity predicts.

Common mistake

Forgetting the Sun's effect. At this level of precision, the simple formula needs corrections: for the Sun's corona, for Earth's air, and for gravity itself.

Where it leads