Question
A probe launched from a spacecraft moves towards the event horizon of a black hole.
a.i. State what is meant by the event horizon of a black hole.[1]
a.ii.The mass of the black hole is $4.0 \times 10^{36} \mathrm{~kg}$. Calculate the Schwarzschild radius of the black hole.[1]
b. The probe is stationary above the event horizon of the black hole in (a). The probe sends a radio pulse every 1.0 seconds (as measured by
clocks on the probe). The spacecraft receives the pulses every 2.0 seconds (as measured by clocks on the spacecraft). Determine the distance of the probe from the centre of the black hole.[3]
▶️Answer/Explanation
Ans:
a.i. the distance from the black hole at which the escape speed is the speed of light
$
\text { a.ii } R_{\mathrm{S}}=« \frac{2 \mathrm{GM}}{\mathrm{c}^2}=\frac{2 \times 6.67 \times 10^{-11} \times 4.0 \times 10^{36}}{9.0 \times 10^{16}}=» 5.9 \times 10^9 « \mathrm{~m} »
$
b. $2=\frac{1}{\sqrt{1-\frac{5.9 \times 10^9}{\mathrm{r}}}}$ rearranged to give $r$
OR
$
\begin{aligned}
& r=1.33 \times 5.9 \times 10^9 \text { «m» } \\
& r=7.9 \times 10^9 « \mathrm{~m} » \checkmark
\end{aligned}
$
Question
It is believed that a non-rotating supermassive black hole is likely to exist near the centre of our galaxy. This black hole has a mass equivalent to 3.6 million times that of the Sun.
a.i. Outline what is meant by the event horizon of a black hole.
a.ii.Calculate the distance of the event horizon of the black hole from its centre.[2]
$
\text { Mass of Sun }=2 \times 10^{30} \mathrm{~kg}
$
b. Star S-2 is in an elliptical orbit around a black hole. The distance of S-2 from the centre of the black hole varies between a few light-hours and
several light-days. A periodic event on S-2 occurs every $5.0 \mathrm{~s}$.[2]
Discuss how the time for the periodic event as measured by an observer on the Earth changes with the orbital position of S-2.
▶️Answer/Explanation
Ans:
a.i. boundary inside which events cannot be communicated to an outside observer
OR
distance/surface at which escape velocity $=c$
OWTTE[1 mark]
a.iimass of black hole $=7.2 \times 10^{36}$ «kg
$
« \frac{2 \mathrm{GM}}{\mathrm{c}^2}=» 1 \times 10^{10} « \mathrm{~m} »
$[2 marks]
b. wherever S-2 is in orbit, time observed is longer than $5.0 \mathrm{~s}$
when closest to the star S-2 periodic time dilated more than when at greatest distance
Justification using formula or time is more dilated in stronger gravitational fields [2 marks]