PHYS320 (Astronomy and astrophysics I) Interstellar medium and formation of protostars

1. (7 pts) The Orion Nebula is a very large nearby molecular cloud. The central region is dominated by four rather bright stars called the Trapezium (the stars with the diffraction spikes in the Hubble Space Telescope image below). The first table below gives the star designation, the apparent magnitude, the observed color index B−V(obs)$B-V\left(obs\right)$, and the Spectral Type. The intrinsic values of color index, absolute magnitude and stellar mass for O and B spectral types is given in the second table. Note that spectral type B0.5 is midway between spectral types B0 and B1.

a) Using the observed apparent (mV$m_V$) and intrinsic absolute (MV$M_V$) magnitudes without correction
for stellar extinction, what are the four distances you would derive for these stars? (2 pts)

b) The observed color indexes in the upper table are considerably larger than the intrinsic color
indexes for the corresponding spectral type in the lower table. Does this mean that the stars
appear redder than they should, or bluer? Which component of the molecular cloud, the gas
or the dust, is the main reason for the color change? (1 pts)

c) What are the four values of color excess (CE$CE$)? For the Orion region, you can use the relation
AV∼6CE. Use this relation to give the corresponding extinction magnitudes, AV$A_V$. (2 pts)

d) Applying the correction for extinction, what are the distances you obtain for the four stars?
Compare with the established distance to the Orion Nebula of 490 pc. (2 pts)

2. (4 pts) Begin with a spherical pure hydrogen molecular cloud (H2) of 8 solar masses (8 Msun), temperature of 10 K, and number density nH2 of 10^10 m-3.

(a) Considering that (mass) density is mass/unit volume. If the density of the cloud is uniform, derive that the radius of the cloud R in terms of Msun, mH, and nH2. Putting in the numbers, what will be the radius of the cloud? Express your answer in AU. (2 pts)

(b) Compare to the Jeans length (equation 12.16 of the text). Will the cloud collapse? (2 pts)

3. (4 pts) If the cloud in Problem 2 collapses,

(a) How long will the collapse take, if given by the free-fall time tff (equation 12.26)? (2 pts)

(b) If the cloud starts with a tiny rotation velocity of 0.5 m/s, how fast, in m/s will it be rotating by the time it reaches a radius of 10^10 m? (2 pts)