Twinkle Twinkle Little Star: The Formation of Stars

(This is the write up of worksheet 10)

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!

JackieNovember 22, 2011 at 1:12 PM
Another result of the last part is that high-mass stars start fusing hydrogen (roughly when they hit the main sequence) long before they have accreted all their mass, whereas low-mass stars have accreted most of their mass already. I wonder if this is a problem for formation of high-mass stars - generally the light from the star will heat up the gas around the star and puff it outwards, making it harder to continue accreting.
JackieNovember 22, 2011 at 1:15 PM
In general, it's a good idea to define your symbols: what is M_c, c_s, etc?

What theorem do you use to derive c_s^2 ~ GM/R?

Sunday, November 13, 2011