Investigating the Schmidt Star Formation Law using numerical simulations
Stephanie J. Bush
Case Western Reserve University
Departments of Physics and Astronomy
Adviser: Dr. Chris Mihos
The Schmidt Star Formation Law was first proposed by Maarten Schmidt in 1959  and states that star formation rate is proportional to the gas density of a cloud raised to some power. We adopt a very generic mathematical representation of this:
(dM/dt)/M = -C*M^n
So the mass the cloud loses as it turns into stars (per unit mass) is proportional to the mass of the cloud to the n.
In order to quantify our understanding of star formation the parameter n needs to be determined.
In order to calculate n, we need to be able to compare the gas density of a galaxy to it's star formation rate. Due to resolution limits, the most common ways of comparing these quantities are by averaging each quantity in concentric rings (e.g. Kennicutt 1989)  , or by averaging each quantity over the entire optical disk of the galaxy (e.g. Kennicutt 1998)  . We can then use several rings in a galaxy or several galaxies as a sample to calculate n.
There are several issues with this method of calculating n that we plan to address in this project. While these methods give us a handle on the Schmidt law on global scales, in order to accurately describe star formation in simulations we need to measure n on smaller scales. Also, when the values given by these methods (in this case Kennicutt 1998) are used in simulations the galaxy's gas depletes much too fast for galaxies to exist with the amounts of gas we measure. (Hernquist, private communication.) One idea to reconcile this is that the local Schmidt law is different enough from the global Schmidt law. So if an individual gas cloud obeys a Schmidt law with one set of parameters, when the properties of the gas clouds are averaged over, we actually measure a different set of parameters.
In order to test this idea we set up a simulation of a galaxy where the star formation follows a Schmidt law with parameters we define, and then "observe" the galaxy to see what parameters we actually measure. Hopefully comparing the two sets of parameters will help us determine why simulations that incorporate the current observed parameters deplete too quickly. In the next few sections I present in detail:
We set up a galaxy to roughly approximate the Milky Way. It follows an exponential density falloff with a scale length of 3 kpc and extends for five scale lengths. On top of this potential we superimposed spiral arms following the prescription in Cox and Gomez (2002)  . We initialize the galaxy with an initial central surface density of 85 M_sun/pc^2, which gives it about 5.1e9 M_sun of gas.
Particles were set up on orbits following a logarithmic potential:
phi = Vc^2/2*ln(Rc^2+R^2)
where Vc is the maximum velocity of the rotation curve and Rc is the galaxy core radius. We set up the galaxy to have a maximum velocity of 220 km/s and a core radius of .5 kpc.
Star formation of each particle follows the Schmidt star formation law:
(dM/dt)/M = -C*M^n
and C is normalized such that dM/dt is 1 M_sun/yr initially. However, there is a considerable amount of evidence that there is a threshold for star formation. At any given radius, a certain amount of gas is needed for it's gravity to overcome the shear in the galaxy tearing it apart in order for it to collapse and form stars. We adopt Martin & Kennicut's (2001)  prescription for the cutoff or critical density: rho_crit = alpha_Q*sigma/(pi*G)*sqrt(2*(v^2/r^2+v/r*dv/dr)) where sigma is the velocity dispersion of the gas, and alpha_Q is an observationally fitted parameter.
For each time step, a star formation rate and mass converted into stars is calculated based on the gas density of the background at the location of each particle. For the next time step, the mass converted to stars is subtracted off the particle's mass. In order for the background to respond correctly, the average mass in each bin (calculated for the azimuthally averaged output) is interpolated over to provide a depleted normalization for the background at any given radius.
We found the best results are given by simulations run with 20000 particles with 400000 times steps to cover 20 Gyrs. We average the results over 40 bins, so each bin is around .4 kpc wide.
We use several plots with the azimuthally averaged data to help us understand how star formation proceeds in the galaxy. Here are the results where the local star formation law incorporates n=1:
With no cut
With hard low cut
With monte carloed cut
With high monte carloed cut
Note, the fifth plot on the edge is not reliable yet.... The red line is just the averaged tau (also plotted in the bottom left plot) at t=0. The black line represents actually where particles deplete by a factor of 1/e, but because of the way it is calculated it has a maximum of 20 Gyrs, which is why it flattens out at the end. The blue line is M/(dm/dt) for a given particle averaged over the particle's orbital time, but at the moment neglects times of no star formation int he average, so needs to be fixed. Once this is working it will give us a good idea of whether the depletion time derived from azimuthally averaged data gives an accurate representation of the depletion time on local scales.
The plot on the upper right is the radial gas distribution. It depletes slowly with time with a form that depends on the form of the cutoff. The galaxy forms stars even when the average mass for a radius is below the cutoff, because the spiral arms still have surface densities above the cutoff. In the outer parts of the galaxy, where even the spiral arms' densities are below the cutoff, there is very little depletion.
The plot on the lower right is the radial star formation distribution. Note that how far out the galaxy forms stars is highly dependent on the cutoff. Of course, when there is no cutoff, the galaxy forms stars at all radii. A hard cutoff severely limits star formation. In the hard cutoff plot, you'll notice that the galaxy only forms stars out to about 12 kpc. This cutoff is actually a factor of two lower than the cutoff proposed by Martin & Kennicutt (2001), because the original cut was so high that the galaxy did not form stars at the solar radius. However, when we monte carlo around that cutoff, the galaxy forms stars at much larger radii. The first monte carlo plot shows that stars form almost over the entire disk. Now we can eliminate our arbitrary factor of two, and creating the second monte carlo plot, where the cutoff is much higher, but the galaxy still forms stars out to 13 kpc.
The cutoff also has a strong affect in the inner regions. In the core of the galaxy, the radial velocity differential is strong enough to make the cut very high in the inner regions, and though plenty of star formation goes on, a reservoir of gas survives.
The upper left plot is the plot we derive the Schmidt law from. When there is no cutoff, the law shows a straight line with a slope of twice the local star formation law. (We confirmed this analytically, it is a result of azimuthal averaging). When the cutoff is added the star formation drops at low and high radii, curving the line and changing n. Eventually, it would be nice to be able to numerically relate the n we observe from the azimuthal averaging to the n we specified locally.
The lower left plot gives the depletion time for a ring, M/(dM/dt), and it shows the behavior we expect: a short timescale in the regions where star formation is high, with the cutoff the highest level of star formation is around 5-7 kpc, depletion times are short. In regions of low star formation, the depletion times are long.
All of these plots show that the star formation cutoff strongly affects the distribution of gas and stars in the galaxy even if it does not strongly affect the globally averaged star formation rates. Martin & Kennicutt's cutoff strongly depletes the middle regions of the galaxy while leaving a core of gas at the center (consistent with star-bursts seen in the interaction and mergers of late type galaxies).
Except for a few minor fixes, the next step in the project is probably to start sph models....