Astrophysics: pressure in a supernova as in explodes

In summary: This is called sleep is good for you! The next morning I redid my calculation and got something in that range - turns out I was accidentally multiplying by Boltzmann constant instead of dividing. :)
  • #1
beee
8
0

Homework Statement


In its early phase, the supernova is an explosion within the star’s own envelope; about half the energy is thermal and half is kinetic. Estimate the ratio of radiation pressure to gas pressure in the star at the moment the shock reaches the surface. (Not in the pre-explosion star!)
Known values: radius = 50R⊙, M = 16M⊙, energy of the explosion = [itex]10^{51}[/itex] ergs, about M = 14M⊙ are ejected; the rest collapse.

Homework Equations


The equations for pressures that I know are:

[itex]P_{rad}[/itex] = a/3 * [itex]T^{4}[/itex]
[itex]P_{gas}[/itex] = ρ/μ * kT

Where μ is mean molecular mass.

I also have [itex]E_{thermal}[/itex] = 3/2 M/μ kT (where M/μ is to estimate the number of particles).

The Attempt at a Solution




I have substituted mass/volume for ρ and divided the two equations by each other getting:
[itex]P_{rad}[/itex]/[itex]P_{gas}[/itex] = 4πaμ[itex]T^{3}[/itex][itex]R^{3}[/itex] / (9Mk)

Evaluating it all I get the ratio to be 3.23 * [itex]10^{-11}[/itex] μ[itex]T^{3}[/itex] (with SI units used throughout, not cgs).

If I use E = [itex]10^{51}[/itex] ergs = [itex]10^{44}[/itex] J given in the setup of the question and equate half of that to [itex]E_{thermal}[/itex] to try to get the value of T, I come up with some terribly small estimates, like [itex]10^{-11}[/itex]K. This is obviously wrong, but I don't quite get what should I change to get the correct result.

Any pointers appreciated.
 
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  • #2
beee said:
If I use E = [itex]10^{51}[/itex] ergs = [itex]10^{44}[/itex] J given in the setup of the question and equate half of that to [itex]E_{thermal}[/itex] to try to get the value of T, I come up with some terribly small estimates, like [itex]10^{-11}[/itex]K. This is obviously wrong, but I don't quite get what should I change to get the correct result.

Any pointers appreciated.

I think you are just calculating it wrong. Let's do a ballpark approximation.

The number of particles N ~ 10^31*10^27 = 10^58
Thermal energy E ~ 2*10^43 J ~ 10^66 K
-> Temperature ~ 10^66K/10^58 =10^8 K.
 
  • #3
clamtrox said:
I think you are just calculating it wrong. Let's do a ballpark approximation.

The number of particles N ~ 10^31*10^27 = 10^58
Thermal energy E ~ 2*10^43 J ~ 10^66 K
-> Temperature ~ 10^66K/10^58 =10^8 K.

This is called sleep is good for you! The next morning I redid my calculation and got something in that range - turns out I was accidentally multiplying by Boltzmann constant instead of dividing. :)
 

Related to Astrophysics: pressure in a supernova as in explodes

1. What is the pressure inside a supernova as it explodes?

The pressure inside a supernova as it explodes can reach up to 10^31 Pascals, which is about 100 billion times greater than the pressure at the center of the Earth.

2. How does the pressure in a supernova contribute to its explosion?

The intense pressure in a supernova is a result of the rapid and violent fusion reactions that occur during the explosion. This pressure helps to counteract the force of gravity and allows the explosion to continue.

3. How is the pressure in a supernova calculated?

The pressure in a supernova is calculated using the ideal gas law, which takes into account the temperature, density, and volume of the gas inside the supernova. Other factors such as radiation pressure and degeneracy pressure may also be considered.

4. Can the pressure in a supernova be simulated in a laboratory?

Currently, the pressure in a supernova cannot be fully simulated in a laboratory setting. However, scientists can use powerful lasers to create conditions similar to those in a supernova and study the resulting pressure and energy release.

5. How does the pressure in a supernova affect the elements produced in the explosion?

The extreme pressure in a supernova is essential for the creation of heavy elements such as gold, silver, and uranium. This pressure allows for the fusion of lighter elements into heavier ones, leading to the formation of these elements in the explosion.

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