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The Shapiro delay

Space and time · level 8, lesson 5 of 6

A question to start

Near the 1976 conjunction, radio signals to the Viking spacecraft came back up to about a quarter of a millisecond later than the distance allowed. Was something wrong with the clocks?

The idea

Irwin Shapiro pointed out that light passing near the Sun should take a little longer than in flat space, because it travels through curved space and time deep in the Sun's gravity. This is the Shapiro delay.

For a radio signal from Earth to Mars and back, passing the Sun at a closest distance \(b\), the extra round-trip time is about

\[\Delta t \approx \frac{4GM_\odot}{c^3}\ln\!\left(\frac{4\,r_E\,r_M}{b^2}\right),\]

where \(r_E\) and \(r_M\) are the distances of Earth and Mars from the Sun. The front part, \(4GM_\odot/c^3\), is only about 19.7 µs. The logarithm makes it grow when the signal skims the Sun: up to about 250 µs.

Analysis of 14 months of Viking ranging data confirmed the prediction to about 0.1%. Later, radio links with the Cassini spacecraft, passing near the Sun, tested general relativity even more closely: general relativity agreed to about 2 parts in 100,000.

signal in the Sun's well:up to about 250 µs lateflat-space path

Worked examples

1. The front part. \(\frac{4 \times 1.327 \times 10^{20}}{(2.998 \times 10^8)^3} = \frac{5.31 \times 10^{20}}{2.69 \times 10^{25}} \approx 1.97 \times 10^{-5}\) s.

2. Skimming the Sun. \(r_E = 1.496 \times 10^{11}\) m, \(r_M = 2.279 \times 10^{11}\) m, \(b = 6.96 \times 10^8\) m. \(\frac{4 r_E r_M}{b^2} \approx 2.8 \times 10^5\), and \(\ln(2.8 \times 10^5) \approx 12.6\). So \(\Delta t \approx 19.7 \times 12.6 \approx 248\) µs.

3. Ten Sun radii out. \(b\) is 10 times bigger, so the ratio is 100 times smaller: \(\ln(2{,}800) \approx 7.9\). \(\Delta t \approx 19.7 \times 7.9 \approx 156\) µs. The delay fades slowly.

Try it

  1. What causes the Shapiro delay?
  2. Why is it largest near solar conjunction?
  3. Convert 250 µs of round-trip delay into an apparent extra distance.
  4. How precisely did the Viking ranging confirm the prediction?
Answers
  1. The signal passes through curved space and time in the Sun's gravity, so it takes longer than in flat space.
  2. The signal then passes closest to the Sun, deepest in its gravity.
  3. \(299{,}792 \times 0.000250 \div 2 \approx 37\) km.
  4. To about 0.1%.

Common mistake

"The signal slows down because of the Sun's gas." The corona does delay radio a little, but that depends on the radio frequency. Using two frequencies lets it be measured and removed. The Shapiro delay is the same at every frequency.

Where it leads