## Sunday, November 13, 2011

### The Formation of Stars

(This is the write up of worksheet 10)

1. The spatial scale of star formation:
• If you let the size of your body represent the size of the star forming complex, how big would the forming stars be?  Let's consider a star forming region (like the Taurus region) that is about 30pc. 1 parsec is roughly 206264 AU, and 1 AU is roughly 100 solar diameters. If we consider the size of a typical star to be roughly that of the sun, a star is 1/30*20626400 in size of the star forming complex. Now let's compare this estimate to the human body. Say the average human body is 1.5 meters; then 1.5/618792000 is roughly 2.42*10^-9 meters. That's on the order of a tenth of the size of a cell membrane! That sure puts things into perspective!
• The average density of a typical star (using our sun as a model) is: $\frac{2\times10^{33}\mathrm{gm}}{\frac{4}{3}\pi(7\times10^{10}\mathrm{cm})^3}\simeq 1.4\mathrm{gm\; cm}^{-3}$ If the Taurus complex contains roughly 3*10^4 solar masses in a radius of 15 parsecs, then the average density of the region is 3.0*10^-23 times less than that of a typical star, which is roughly 1.4*10^-23 g/cm^3.

3. Proto-stars and the pre-main sequence

• The rate at which mass accumulates onto a forming star is                                                 $\dot{M}\sim \frac{M_c}{t_{dyn}}$         $t_{dyn}= \frac{R}{c_s}$   $c_s^2\sim\frac{GM_c}{R}$                                                           rearranging these equations, we find that                                                                        $\dot{M}\simeq \frac{c_s^3}{G}$
• To find the time it would take to build up stars of a certain mass (1 solar mass and 30 solar masses in this problem), we integrate the accretion rate with respect to time.                                                                    $\int \dot{M}dt = \frac{c_s^3}{G}t$                                                                                    using cgs units where G=6.8*10^-8 cm^3/g/s^2, Msun = 2*10^33 g, and the speed of sound = 2.5*10^4cm/s , we find that it takes 8.7*10^12 seconds to build up a 1 solar mass star and 2.6*10^14 seconds to build up a 30 solar-mass star. In more familiar units, it takes roughly 2.8*10^5 years to build up a 1 solar-mass star and 8.3*10^6 years to build up a 30 solar mass star.
• The Kelvin-Helmholtz time scale is given by                               $t_{KH}=\frac{\Delta E_g}{L_{\odot}}$  $\Delta E_g = \frac{3GM^2}{5R}$
In this problem, we are dealing with a 1 solar mass star (L ~ 100 solar luminosities) and a 30 solar mass star (L ~ 10^5 solar luminosities). Using the above equations and the fact that mass scales roughly as radius, we find that the Kelvin-Helmholtz time scale for the 1 solar mass star is 5.8*10^12 seconds ~ 1.8*10^5 years and the time scale for the 30 solar mass star is 1.7*10^11 seconds ~ 5.5*10^4 years. These numbers are an order of magnitude smaller than the accretion time scales.
• The results from the previous part suggest that larger stars reach their main sequence faster than smaller stars. In general, larger stars have much shorter life-spans than smaller stars.
I would like to thank Juliette and Tommy for working with me!