Explain cosmic distances to a child

Question

I seem to have lost all my sense for simple calculations. I'd like to explain to my son how long cosmic distances are. As an example:

Our Sun has a diameter of $1392684\;\text{km}$. The distance to our next star Alpha Centauri is $4.153\cdot10^{13}\;\text{km}$. Now, if you pretend the Sun to be as large as a pinhead ($1\;\text{mm}$), the distance to Alpha Centauri would be... $6.793\;\text{km}$?

Sorry for this super-trivial question, but I am just not sure enough if my result is correct or a couple of digits wrong. Am I right?

EDIT: Thank you Claude for noticing my second error - the Sun has a diameter of $1392684\;\text{km}$, not $1392684000\;\text{km}$ as I wrote earlier.

Answer 1

The sun has a diameter of

$$d_{\rm Sun} = 1.39 \times 10^6 \textrm{ km}$$

The distance to Alpha Centauri (in km) is

$$d_{\rm AC} = 4.13 \times 10^{13} \textrm{ km}$$

The ratio of these quantities is

$$\frac{d_{\rm AC}}{d_{\rm Sun}} = \frac{4.13 \times 10^{13}}{1.39\times 10^6} = 2.97 \times 10^7$$

Therefore, if you had a model where the diameter of the sun was 1 mm, the distance to Alpha Centauri in the model would be 2.97 x 10^7 times greater, i.e.

$$2.97 \times 10^7 \textrm{ mm} = 29.7 \textrm{ km}$$

If you took the pinhead representing the Sun and stood next to the Washington Monument in Washington, DC, then Alpha Centauri would be about halfway to Baltimore.

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Edit, more appropriate since you live in Austria! If you stood in the center of Vienna, then Alpha Centauri would be about halfway to Bratislava.

The Sun:

Alpha Centauri:

Answer 2

Going to powers will simplify. If, as you gave, the diameter of the sun was $1.392684\times 10^9$ km and the distance to Alpha Centauri is $4.153\times 10^{13}$ km, this makes a ratio of $29820$. So, if the diameter of the sun was $1$ mm, the distance to Alpha Centauri would be $29820$ mm, that is to say $29.820$ meters.

The problem is that the diameter of the sun is $1.392684\times 10^6$ km; the same reasoning will then bring you to $29.820$ kilometers.