Question

Take an idealized galaxy made of 1011 sun-like stars distributed in a disk that is 50 kpc in diameter and 1 kpc thick. We wish to build (i.e. with our hands) a scale model of the galaxy substituting a grain of sand and shrinking each star proportionally. (Assume R∗ = 7 × 108 m and Rsand = 1 mm).

a) What is the size of the model galaxy’s disk? Practically speaking is this a good scale to use for our model?

b) What is the average distance in km between stars (grains of sand) in our scale model?

c) If you collected all the sand into one place, what is the smallest volume it could fill? How does this compare with the volume of our classroom? (guestimate!)

d) Given what you have just learned from our scale model, do you expect stars to collide when two galaxies merge?

please explain answers so I can learn

Answer 1

Given:

Number of stars: 10¹¹

Diameter of galaxy= 50 kpc = 50*1000*3*10¹³ km = 15*10¹⁷ km

Thickness of galaxy = 1 kpc = 3 * 10¹³ km

Shrinking ratio =  $\dpi{100} \fn_phv \small \frac{Radius of sand}{Radius of Star}$  

= 1*10^-3 m/7*10⁸ m = 1/(7 * 10¹¹ )

a) Size of model galaxy disc = size of actual galaxy * compression ratio

= 15 * 10¹⁷ / 7 * 10¹¹ km = 2 * 10⁶ km

We can see that the size of model comes out to be 2 million kms. As a comparison, earth's diameter is mere 12,600 kms. So we see that the model is impossible to make, and we need much smaller scale. This shows how vast our galaxy is, and how much tiny place we occupy in this galaxy.

b) Avg distance between stars in real galaxy = 5 * 10¹³ kms

For model, average distance = 5 * 10¹³ / 7 * 10¹¹ km = 49 kms

So we see that we will have to place each sand particles nearly 50 km apart. This shows how much space is there between two stars.

c) Assuming 10¹¹ stars of 1mm radius each, the total volume will be :

10¹¹ * (4/3) * $\dpi{100} \fn_phv \small \Pi$ * (10^-3)³ = 400 m³  

Average volume of classroom = 125 m³

We see that the volume of sands put together is more than double the volume of an average classroom.

d) We see that the distance of separation between two stars is extremely large compared to their radius. Imagine two sand particles separated in space by 50 kms! So we can safely say that when two galaxies merge, the stars will not collide, as there is so much empty space between them (there may be very few rare case of collision).