Density at the centre of the sun

In summary, the question is asking to calculate the total gravitational potential energy of a gravitating sphere with a given density profile. To do so, an expression for the center density \rho_{center} in terms of the radius of the star and its mass must be derived. The problem can be solved by constructing a function M(r) using the given density profile and integrating it over all radii. However, the question specifically asks for an expression for \rho_{center}, so the value must be calculated from scratch rather than using a given value. This can be done by setting r=0 in the expression, but care must be taken as any attempt may result in infinity due to r being in the denominator.
  • #1
maximus123
50
0
Hello, here is the question I have to answer;

Calculate the total gravitational potential energy [itex]U[/itex] of a gravitating sphere of mass [itex]M[/itex] with a density profile [itex]\rho(r)[/itex] given by

[itex]\rho(r)=\rho_{center}\left(1-\frac{r}{R_{star}}\right)[/itex]​

where [itex]R_{star}[/itex] is the radius of the star and [itex]\rho_{center}[/itex] is the density at [itex]r=0[/itex]. First give an expression for the center density [itex]\rho_{center}[/itex] in terms of [itex]R_{star}[/itex] and [itex]M[/itex], then compute a value for the sun. Calculate the total gravitational potential energy of the sun.
I am aware that the gravitational energy of one layer of thickness dr is
[itex]dU=-\frac{GM(r)dm}{r}[/itex]​
and that ultimately I will have to integrate this over all radii but I am unclear about the expression for [itex]\rho_{center}[/itex]. The only thing that springs to mind is
[itex]\rho=\frac{M}{\frac{4}{3}\pi R^3}[/itex]​
but this must be for an average density over the whole star. Can anyone point me in the direction of how to establish an expression for [itex]\rho_{center}[/itex]?

Thanks a lot
 
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  • #2
It says in the problem that it is the density at the center ie r=0 its just a constant scalar.

so you must construct a function M(r) using p(r) for the shell.
 
  • #3
However the question says "give an expression for [itex]\rho_{center}[/itex], so presumably I have to calculate the value of [itex]\rho_{center}[/itex] from scratch rather than looking it up. For example I know that the value for the center density quoted from many sources is [itex]1.622\times10^5\textrm{ kg m}^{-3}[/itex] however it is clear from the question I cannot simply use this value but I need to form an expression and then set r=0, but anything I try always ends up with r in the denominator thus resulting in infinity.
 

Related to Density at the centre of the sun

1. What is the density at the centre of the sun?

The density at the centre of the sun is estimated to be around 150 g/cm³. This is about 150 times the density of water.

2. How is the density at the centre of the sun measured?

The density at the centre of the sun is not measured directly, as it is impossible to reach the core of the sun. Instead, scientists use mathematical models and observations of the sun's surface to estimate the density.

3. How does the density at the centre of the sun compare to the density of other objects?

The density at the centre of the sun is much higher than the average density of planets in our solar system. It is also denser than most stars in the universe.

4. What factors affect the density at the centre of the sun?

The density at the centre of the sun is affected by various factors, including the temperature, pressure, and composition of the sun's core. As the temperature and pressure increase towards the centre, the density also increases.

5. Why is the density at the centre of the sun important?

The density at the centre of the sun is important because it plays a crucial role in the sun's energy production through nuclear fusion. The high density and temperature at the core allow for hydrogen atoms to fuse together, releasing vast amounts of energy that sustain the sun's heat and light.

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