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I am experimenting with multi body simulations in astronomy ($N$-body algorithms). It is customary to give the results in dimensionless units.

One of them is a length unit, and the virial radius is often used to normalize the values. It is used for star clusters, galaxies and galaxy clusters. I read about it, but I am confused about the cosmological background.

Is it possible to calculate the virial radius of a system of point masses given their positions and velocities? Can we get the virial radius for the Solar System?

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The "virial radius" is usually used in the context of cosmological simulations, on the scale of galaxies or clusters of galaxies. It's not usually used for the Solar System or star clusters, and indeed, the usual definition would fail. Note that you shouldn't read too much into the name "virial". While there's a link to the virial theorem if you look hard enough, it's not that obvious and not that important.

To make matters worse, there are many definitions of radii that different authors call "the virial radius". In general, though, it is defined as a radius within which the average density is equal to some characteristic value. One common definition is $R_{\rm vir}=R_{200{\rm c}}$, where the characteristic density is $200$ times the critical density of the Universe, $3H^2/8\pi G$, where $H$ is Hubble's constant and $G$ is Newton's constant. The number $200$ comes from a calculation of the virialization of a spherical top-hat overdensity which can be found in Peebles (1980). The exact result for a critical universe is actually $18\pi^2\approx177$, but since the calculation relies on some approximate assumptions anyway, it's become common to just round the value to $200$. Notice that $H$ changes with time, so the characteristic density also changes with time, meaning that using this definition, the virial radius will change with time even for a static system. This is sometimes called "pseudo-evolution".

Sometimes the characteristic density is defined in terms of the average density of the Universe instead of its critical density. In the real Universe it seems that these are almost the same, but there are cosmological models in which this is not true.

In practice, to measure the virial radius for a collection of particles, you pick a centre, then start adding up the mass within a radius $r$ of that centre point and dividing by the volume, increasing $r$ until the density drops to the desired characteristic value.

As a bit of trivia, it is straightforward (but not very useful) to calculate the virial radius of the Solar System, by solving for $R_{200{\rm c}}$ in: $$\frac{3{\rm M}_\odot}{4\pi R_{200{\rm c}}^3}=200\frac{3H^2}{8\pi G}$$ This assumes that all the mass in the Solar System is in the Sun, which is nearly true. Putting in typical values I get $R_{200{\rm c}}\approx6\times10^{17}\,{\rm m}\approx4\times10^6\,{\rm AU}\approx20\,{\rm pc}$. This is big enough to enclose hundreds of nearby stars, which is why it's not a terribly useful number to work with.

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