Scale and Distance Calculations

Use $1\mathrm{AU}=1.50\times 10^8\mathrm{km}$ and $1\mathrm{ly}=9.46\times 10^{12}\mathrm{km}$.

1. Jupiter is $5.20\mathrm{AU}$ from the Sun. How many kilometres?

Answer

$5.20\times 1.50\times 10^8=7.80\times 10^8\mathrm{km}$

2. Light takes 8 min 20 s to reach Earth. Show that this gives 1 AU.

Answer

$8\min 20\mathrm{s}=500\mathrm{s}$, and $d=vt=(3.00\times 10^{5}km/s)(500\mathrm{s})=1.50\times 10^8\mathrm{km}$ ✓

3. Proxima Centauri is $4.24\mathrm{ly}$ away. Convert to kilometres, then to AU.

Answer

$4.24\times 9.46\times 10^{12}=4.01\times 10^{13}\mathrm{km}$

Divided by 1 AU: about $268000\mathrm{AU}$. The nearest star is a quarter of a million times further away than the Sun.

4. Voyager 1 travels about $17km/s$. How long to reach Proxima Centauri?

Answer

$t=\frac{4.01\times 10^{13}}{17}\approx 2.4\times 10^{12}\mathrm{s}$, which is roughly 75 000 years.

Voyager is not headed there — but the number is the point. Our fastest departing object would need longer than recorded history, many times over.

5. Andromeda is 2.5 million light-years away. How old is the light you see?

Answer

2.5 million years old. It left before Homo sapiens existed; early humans were making the first stone tools. Every telescope is a time machine, and it only points one way.

6. On a model where the Sun is a 24 cm basketball, how far away is Neptune ($4.50\times 10^9\mathrm{km}$)?

Answer

The scale factor is $\frac{0.24\mathrm{m}}{1.39\times 10^9\mathrm{m}}$, giving about $777\mathrm{m}$ — most of a kilometre from the basketball.

Reference: Distances in Space