Shapiro Delay

Shapiro Delay

In 1964, Irwin Shapiro predicted a fourth test of GR: radar signals passing close to a massive body should arrive slightly later than flat-space predictions, because time runs slower in a gravitational field. This is the Shapiro time delay (gravitational time delay).

For a signal travelling from distance $r_1$ to $r_2$ past a mass $M$ at closest approach $b$ (where $r_1, r_2 \gg b$), the excess travel time is approximately:

$\Delta t = \frac{2GM}{c^3} \ln\!\left(\frac{4 r_1 r_2}{b^2}\right)$

Earth–Mars Radar Test

Shapiro and collaborators verified this prediction in 1968–1971 by bouncing radar pulses off Mars and Venus as they passed behind the Sun. For a signal from Earth to Mars grazing the Sun:

This gives a one-way delay of about 124 microseconds, confirmed to better than 0.1% precision.

Round-Trip and Microseconds

For a radar echo (round trip), the total delay is $2 \Delta t$. It is convenient to express the delay in microseconds ($\mu$s) since the delays are tiny fractions of a second.

Your Task

Implement three functions. All physical constants must be defined inside each function body.

• shapiro_delay(M, r1, r2, b) — one-way delay in seconds: $(2GM/c^3) \ln(4 r_1 r_2 / b^2)$
• shapiro_delay_round_trip(M, r1, r2, b) — round-trip delay in seconds: $2 \times \Delta t$
• shapiro_delay_us(M, r1, r2, b) — one-way delay in microseconds: $\Delta t \times 10^6$