Need help calculating the size of the Stromgren sphere

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SUMMARY

The discussion centers on calculating the size of a Stromgren sphere, which is defined as the region around a star where its photons ionize surrounding gas. The key equation involves the number densities of electrons (ne) and protons (np), which are equal in a fully ionized sphere. The participant seeks clarification on rewriting the equation as ne² and how to apply integrals to determine the number density, specifically aiming for a value of 10³/cm³. Additionally, they inquire about the concept of radiative equilibrium and its calculation.

PREREQUISITES
  • Understanding of ionization processes in astrophysics
  • Familiarity with differential equations and integrals
  • Knowledge of number density concepts in plasma physics
  • Basic principles of radiative equilibrium
NEXT STEPS
  • Study the derivation of the Stromgren sphere radius formula
  • Learn about the application of integrals in calculating number densities
  • Research the concept of radiative equilibrium in astrophysical contexts
  • Explore the relationship between ionization and temperature in stellar environments
USEFUL FOR

Astronomy students, astrophysicists, and anyone interested in the dynamics of ionized gas around stars will benefit from this discussion.

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Homework Statement



A stromgren sphere is the sphere surrounding a star where the photons from the star can ionize the gas. So the radius of the sphere is the extent from the star in which the gas is ionized. For a fully ionized (idealized, for this question) sphere of gas, the number of protons would equal the number of electrons in the sphere.

(Please look at the attached image to see the full question, it has a large equation that would be very difficult to type out)

Homework Equations


The Attempt at a Solution



The equation includes nenp, and these two quantities should be equal, so can I just rewrite it as ne2? I figure I will do an integral to find the number of electrons in time, but how can I apply that to the number density? I am assuming that ne and np stand for number of electrons and protons, so how can I calculate that to work out to 10^3/cm^3?

Or is my technique crrect at all? Would my integral be definite or indefinite? And since the temperature term doesn't deal with ne or t, can I pull it out and be left with a simple differential equation?

Also, what is radiative equilibrium and how would I calculate it?
 

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