# What is a Plummer Sphere Exactly?

#### sderamus

I assume it is a sphere with "handles" stretching out from the side - somewhat like a galaxy as that is where I have seen the term. I searched in vain in old Physics texts but could not find what it is exactly.

TIA

Sterling

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#### SpaceTiger

Staff Emeritus
Gold Member
A plummer sphere is just a particular model for density/gravitational potential distribution in a spherical halo of matter. The density distribution is given by:

$$\rho(r)=\frac{3M}{4\pi a^3}\frac{1}{(1+\frac{r^2}{a^2})^{5/2}}$$

The gravitational potential is

$$\Phi(r)=\frac{-GM}{\sqrt{r^2+a^2}}$$

The plummer potential is often used in N-body simulations to "soften" gravity at small distance scales. This is needed to prevent the point particles (which are used as an approximation to extended objects) from scattering too strongly off of one another on close approach.

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#### sderamus

SpaceTiger said:
A plummer sphere is just a particular model for density/gravitational potential distribution in a spherical halo of matter. The density distribution is given by:

$$\rho(r)=\frac{3M}{4\pi a^3}\frac{1}{(1+\frac{r^2}{a^2})^{5/2}}$$

The gravitational potential is

$$\Phi(r)=\frac{-GM}{\sqrt{r^2+a^2}}$$

The plummer potential is often used in N-body simulations to "soften" gravity at small distance scales. This is needed to prevent the point particles (which are used as an approximation to extended objects) from scattering too strongly off of one another on close approach.
Thanks, Tiger. I saw the gravitational potential equation in the text. But what is "a" exactly? Is it the radius of the halo?

TIA

Sterling

#### SpaceTiger

Staff Emeritus
Gold Member
sderamus said:
Thanks, Tiger. I saw the gravitational potential equation in the text. But what is "a" exactly? Is it the radius of the halo?
Just an arbitrary parameter in the Plummer formula. It can be thought of roughly as the radius of the halo, since the potential approaches that of a point mass beyond that radius. In the context of N-Body simulations, "a" will be called the "softening length".

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