Credit: www.wikipedia.com
In this week of Ay20, we learn about the mechanism of how a star is formed. Picture above is an artist's interpretation of the stage while a star is formed. The proto-star sits in the middle surrounded by a cloud of mass accreted into the star forming the accretion disk. Two yellow jets of mass are excreted from the star to maintain the angular momentum. Here, we solve problem number 4 on the worksheet "The Formation of Stars" located here.
where M_c is the mass of the core and t_dyn is the dynamical time, or the time it takes a pressure wave (sound wave) to traverse the cloud. The dynamical time can be written as,
where c_s is the isothermal sound speed and R_J is called the Jeans' length. If the radius of the cloud is larger than the Jeans' lenght, the cloud will undergo the gravitational collapse. Otherwise, The cloud would undergo the stable oscillations. The isothermal sound speed, c_s, can be approximated, to oder of magnitude, as
or write M_c in term of c_s as,
Now, we substitute in known value into the mass accretion rate equation.
The Jeans' length can be written as,
where rho_bar is the average density of the cloud. It can be expressed as
Again, we substitute back into the mass accretion rate equation.
But, we want to express it independent of mass and radius. Thus, we substitute in the expression of the isothermal sound speed. Finally, we get the mass accretion rate equation as desired.
(b) We try to use our accretion rate for mass one solar mass and 30 solar mass. We use c_s to be 0.25 km/s throughout corresponding to a molecular gas at 20 K.
For a cloud of one solar mass,
For a cloud of 30 solar masses,
James Hopwood Jeans
Credit: www.wikipedia.org
William Thomson, 1st Baron Kelvin
Credit: www.wikipedia.org
Hermann von Helmholtz
Credit: www.wikipedia.org
Three pictures above are people who are named after for the "Jeans' length" and "Kelvin-Helmholtz timescale." I just wonder how they looked like.
Let's continue with our work.
(c) Once a protostar forms at the center of gravity, it will contract on a Kelvin-Helmholtz time. It is the gravitational potential energy divided by the luminosity. The gravitational potential energy of a sphere of mass M and radius R can be expressed as,
As Jackie pointed out, should we use the radius of the initial or the final radius? Since the gravitational potential energy is inverse of radius and the initial radius is more than 100 times of the final radius, the final radius dominates. But how could we find the final radius then?
We assume that the star contracts into a star in a main sequence. Our sun is a also a typical main sequence star. The mass of a star is scale as its radius. Thus, a star with mass of one solar mass has one solar radius. And a star with mass of 30 solar mass has radius of 30 solar radius.
For a star of 1 solar mass with 100 solar luminosity,
For a star of 30 solar mass with 10^5 solar luminosity,
(d) The Kelvin-Helmholtz time shows the time when all gravitational potential energy is dissipated by the luminosity. Thus, after this time is ended, a star starts its nuclear reaction. As you can see, t_KH is relatively "short"so it is pretty hard to observe.
Acknowledgement
We thank Wikipedia for pictures of Jeans, Kelvin and Helmholtz. Also, we thank WolframAlpha for doing calculations which simplify many steps for us. We thank Jackie for pointing out the radius in gravitational potential energy. We also thank Prof. Johnson for the assumption that our stars in the question are in the main sequences. All equations here are made by online LaTex.
Mee! I love how you link to your collaborators' blogs and the worksheet.
ReplyDeleteI enjoyed the pictures of Jeans, Kelvin, & Helmholtz. They look so serious! (Which I think was the style for photos back then.) But then I imagine them at a chalkboard (or whatever they did science on back then) having fun talking their way through these problems, just the way I do, and I feel oddly connected to them.
Another thing to note: how do the timescales for accretion and contraction (Kelvin-Helmholtz) compare? Are high-mass stars still accreting when they ignite nuclear fusion?