## Tuesday, October 25, 2011

### Neutron Stars and Their Equations of State

For my summer research with LIGO, I worked a lot with neutron stars.  The first portion of my project was making a mass-radius plot of different proposed equations of state. An equation of state provides the relationship between the thermodynamic variables of pressure and energy density.

I worked with/helped develop a TOV solver (which is a code in C) that makes model neutron stars (i.e. given a  starting central density, outputs the properties of the resulting star). We're most interested in the masses and radii of these stars.

There were two types of equations of state: analytic and tabular. The analytic equation of state was given by the Tolman-Oppenheimer-Volkoff (TOV) equation:

$\frac{dP(r)}{dr}=-\frac{G}{r^2}\left[\rho(r)+\frac{P(r)}{c^2}\right]\left[M(r)+4\pi r^3 \frac{P(r)}{c^2}\right]\left[1-\frac{2GM(r)}{c^2r}\right]^{-1} \;$

It looks really scary... but luckily the program solved it, so I didn't have to :] The equation basically describes a spherically symmetric, isotropic star in static gravitational equilibrium.

The other equations of state came in the form of density-pressure tables from different researchers in the field. I wrote a function that interpolated the pressure-density relationship so that the code could use it instead of the TOV equation to create a model neutron star.

I also had to look at the constraints for the mass-radius plot.

Minimum Maximal Mass Constraint
The first constraint is based on the observation of a 2 solar-mass neutron star. Any equation of
state that does not predict a mass that reaches two solar masses is disproved by obesrvation.

General Relativity Constraint
The GR constraint requires R > 2GM/c2  because neutron stars must have a radius that
is greater than the Schwarzschild Radius. Any proposed neutron star with a radius below
the threshold would in reality be a black hole.

Finite Pressure Constraint
The finite pressure constraint, according to Lattimer and Prakash (2007) , requires
R >(9/4)GM/c2.

Causality Constraint
To obtain the causal constraint, we recreated the procedures of Koranda et al. (1997) ,
and Lattimer and Prakash (2007). To avoid a supraluminous equation of state in which
the speed of sound is faster than the speed of light, it is necessary that

$\frac{dp}{d\rho}\leq c^2$
.
Rotational Constraint
For a fully relativistic star, the rotational limit can be expressed as
$\small R_{sph} < 10.4 \bigg(\cfrac{1000Hz}{\nu}\bigg)^{2/3}\bigg(\cfrac{M_{\mathrm{sph}}}{M_{\odot}}\bigg)^{1/3} km$

The final mass-radius plot  that includes all of the EOS that I worked with looked like this:

1. So do the constants define limits that tell you what type of star it is?

Also, what are all the colored lines on your plot? Are those different stars, or systems?

This looks really interesting!! You should post another entry telling us more about your project :)

2. The constraints don't necessarily define the type of star. They're more for defining physical limits/possibilities of neutron stars. Other stars are considerably less dense, so their mass-radius curves would be shifted much farther to the right. (Consider that for a 1-solar mass neutron star, the radius is somewhere in the range of 9-15km.)

The colored lines represent the mass-radius relationship of neutron stars predicted by a corresponding equation of state.

3. This project looks really fun! Who did you work with? We were supposed to construct a plot like this (called a Lattimer plot, I think), but with only the TOV equation of state, on a homework assignment spring term but it was taking forever and I wasn't sure how to come up with the different constraints, so I ended up skipping it. I wish I had this blog entry back then...

4. Also, does the TOV equation look similar to something we studied in class? What happens to the TOV equation in the non-relativistic limit?

Non-relativistic means:

P(r) / (rho(r) * c^2) << 1 (this is equivalent to sound speed << light speed, which is true when the average energy per particle is much less than the rest energy of the particles)

r >> 2*G*M(r)/c^2 (2*G*M/c^2 = r_s is known as the Schwarzschild radius and is the radius to which you would have to compress mass M to have a black hole. Neutron stars are only a few times bigger than the Schwarzschild radius so the effect of general relativity becomes very important here)

5. Thank you for sharing this information.
It will really helpful to solve my confusion

Process $Chemical Engineering 6. nr: You're very welcome! If you have any more questions, I'd love to chat more on the subject. Jackie: In retrospect, this class helps me understand my summer research a lot more. The TOV equation looks a lot like the equation for Hydrostatic Equilibrium :] I worked with Christian Ott and Jeff Kaplan last summer. It was really awesome! When I started my research, I had never seen the equations of stellar structure, so that made things a lot more confusing. I feel a lot more comfortable with the material now, and I think neutron stars are the coolest :] 7. Thank you for sharing this information. It will really helpful to solve my confusion Process$ Chemical Engineering

8. Anybody has Wolfram Mathematica code for Mass-Radius Plot?
Would be extremely grateful!