Question

1. Formation of “Structure”. Galaxies form out of the primordial fluctuations that we observe on the CMB, and stars form in galaxies. Here, we investigate a few key processes for galaxy and star formation, using order of magnitude estimates.

(a) Typical spacing between galaxies. Take the average density in the universe today (~ 30% of the critical density) and calculate the volumn needed if one wants to collect enough mass to build the Milky Way (a total mass of ~ 10^42 kg). If galaxies are roughly equally spaced in the universe, this will yield the typical distance between Milky-way type galaxies. For context, our closest twin, the Andromeda galaxy, sits at 2.5 million light-years away from us. Comparing your result against this value and what conclusion can you infer about the environment density in which the Milky Way galaxy form?

(b) Merging is important for building up galaxies. In class, it is stated that galaxies collide, but stars don’t (usually) do. This is because galaxies are separated by a distance that is great (typically 10^3 kpc), but not much greater than their physical sizes (typically 100 kpc), while stars are separated by a distance (typically 1 pc) that is much, much greater than their physical sizes (typically 10^11 cm). Assume both galaxies and stars are running on random straight path (i.e., not bent by the gravity of other bodies) with a speed of 500 km/s. How frequent do stars hit each other, and how frequent do galaxies hit each other? (express both in unit of years) (Hint: arrange stars/galaxies randomly in space, the mean distance for a collision is l ~ 1/(n*pi*R^2), where n is the number density and R the physical size. You can easily derive this relation by asking the following question: assume all objects are at distance l, how large does l have to be before these objects cover up the entire sky (and therefore the chance of collision is unity)?)

(c) One of the biggest problems in forming stars is the shedding of angular momentum. We examine this issue here. The mass density of a typical molecular cloud (out of which stars form) is ~ 10^-22 g/ cm3, while the mass density inside the Sun is ~ 1g/ cm^3. This means the material that formed the Sun was collected from a region 10^22 times larger in volumn than the Sun. Let this original cloud be spherical and have the same density distribution as that inside the Sun – so that the momentum of inertia I = fMR^2 has the same constant f. If the cloud rotates with a speed v, how small v has to be so that the resultant Sun spins at below its break-up velocity? (Hint: break-up velocity means the star is rotating so fast that its surface rotational velocity becomes comparable to the escape velocity from the star.) For reference, molecular clouds are typically moving and spinning with velocities of order 1 km/ s. So how much of the original angular momentum has to be shed before a star can be born?

(d) Nuclear fusion and the lifetime of the Sun. The mass of a helium nucleus is 6.6447x10^-27 kg, while that of a hydrogen nucleus is 1.6737x10^-27 kg. So when four hydrogen nucleus fuse (in a long chain) to eventually form a helium nucleus, some amount of mass is missing. This missing mass is the source of the Sun’s energy. Now assume that the Sun turns all its hydrogen (~ 75% of its total mass) into helium, and that it continues to shine at its current luminosity (3.84 x 10^26 Watts), how long can it it shine?

Answer 1

1. (a) The critical density is  

Then current density of universewill be  

average length can be assumed to be cube root of volume.

or almost 9 million Light years

The Length is an average length and varies with the galaxy in question. Andromeda is 2.5 million light years away in the same order of magnitude of distance as the average distance between galaxies.

(b)

=>   

The frequency is

  

For galaxies the number density    

500 km = 500 * 3.24 * 10^-27 kpc = 1.62 * 10^-24 kpc.   

for stars the number density,

The average size of star = 10¹¹ cm = 10⁷ km = 3.24 * 10^-24 pc

n = 3.24 * 10^-24 ; R = 1.62 * 10^-24 pc

   is a very small number. Thus we can say that during collision of galaxies, stars don't collide.