Calculate Gravitational Self-Energy of the Sun

In summary, the gravitational self energy of the sun is calculated by integrating the energy needed to take each layer of the sun from its surface to infinity. This is done by considering the mass of each layer and the inner sphere, and then integrating from the outermost layer to the radius of the sun. This results in an answer of \frac{3}{5} G \frac{M^2}{R}.
  • #1
stunner5000pt
1,461
2
Calculate the gravitational self energy of the sun (without using numbers)

Textbook says that this is the nergy needed to take every particle from the sun's surface to infinity

there are an infinite number of particles on the sun ...
would there be some sort of integral that relates the energy needed to take one particle from the sun's surface ?

something like this - for one particle mass i, energy needed ot take it out from the sun's surface [tex] U_{i}(r) = -G \frac{M_{s} m_{i}}{r_{si}} [/tex]
but since this is a 3D object how would one go about setting up an integral like this??
 
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  • #2
First, let's assume that the sun density [tex]\rho[/tex] is constant.

then, let's slice the sun from the center. If we add a layer of [tex]d R[/tex], at a distance [tex]R[/tex] from the center, then the layer mass will be [tex]dM_R=4\pi \rho R^2 dR [/tex]. Now we need to find the gravitational energy between that layer and the outer shell. So let's slice the outer shell as well: [tex] dU_R=-GM_R\int{ \frac{4 \pi \rho r^2 dr}{r}}[/tex], the integration from [tex]R[/tex] to [tex]R_0[/tex], the sun's radius. After that we need to find out [tex]\int {dU_R}[/tex].
 
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  • #3
There is gravitational energy between the layer and the inner sphere (at a particular R). The mass of the inner sphere is just (4/3)pi*R^3*density

If we integrate [tex]GM_{layer}M_{innersphere}/R[/tex] over all R from 0 to the radius of the sun... I think we get the answer.

EDIT: Shyboy's method is also exactly right. I got the same answer both ways. Apologies.
 
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  • #4
what is
[tex] \int dU_{R} [/tex]

what would the integration variable be then??

p.s. The answer is supposed to be [tex] \frac{3}{5} G \frac{M^2}{R} [/tex]

did you guys get that??
 
  • #5
looks like so. I missed [tex]d[/tex] before [tex]M[/tex]. It should be [tex]dM_R[/tex] instead of [tex]M_R[/tex]. And I wish I can put limits in the integral but dont
know how (to lasy to find it) :(
 
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  • #6
int_{lower bound}^{upper bound}

ex:

[tex]\int_a^{b-\epsilon_0}[/tex]
 
  • #7
stunner5000pt said:
what is
[tex] \int dU_{R} [/tex]

what would the integration variable be then??

p.s. The answer is supposed to be [tex] \frac{3}{5} G \frac{M^2}{R} [/tex]

did you guys get that??

Yes. Here's what I did:

[tex]M_{layer} = 4\pi r^2\rho dr[/tex]

[tex]M_{inner} = \frac{4}{3}\pi r^3\rho[/tex]

Remove the layers of the sun away to infinity starting with the outermost layer.

Energy required to take a single layer out to infinity=
[tex]0 - (- \frac{GM_{layer}M_{inner}}{r})=[/tex]
[tex]\frac{16G\pi^2 \rho^2 r^4 dr}{3}[/tex]

If we integrate this from r=0 to r=R (giving the energy to take away all the layers), we get
[tex]\frac{16\rho^2 \pi^2 R^5}{15}[/tex]

If we use [tex]M=(4/3)\pi R^3\rho[/tex], solve for [tex]\rho[/tex] and plug into the above... we get
[tex] \frac{3}{5} G \frac{M^2}{R} [/tex]
 
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Related to Calculate Gravitational Self-Energy of the Sun

1. How is the gravitational self-energy of the Sun calculated?

The gravitational self-energy of the Sun can be calculated using the formula E = -3/5 * (G * M^2)/R, where E is the self-energy, G is the gravitational constant, M is the mass of the Sun, and R is the radius of the Sun.

2. What is the significance of calculating the gravitational self-energy of the Sun?

Calculating the gravitational self-energy of the Sun allows scientists to better understand the internal structure and dynamics of the Sun. It also provides insight into the Sun's role in the gravitational interactions within our solar system.

3. How does the gravitational self-energy of the Sun compare to its total energy?

The gravitational self-energy of the Sun is only a small fraction of its total energy. The majority of the Sun's energy comes from nuclear fusion reactions in its core.

4. Can the gravitational self-energy of the Sun change over time?

Yes, the gravitational self-energy of the Sun can change over time as the Sun's mass and radius change. This can occur due to processes such as nuclear fusion and mass loss through solar wind.

5. How does the gravitational self-energy of the Sun affect other objects in the solar system?

The gravitational self-energy of the Sun contributes to the overall gravitational pull of the Sun, which affects the orbits and movements of other objects in the solar system. This includes the planets, moons, and other celestial bodies.

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