Question

The Eddington limit applies to stars as well as to accreting black holes, and places an upper limit on their mass. Using the mass-luminosity relation for high-mass stars: L/L. = 1.02(M/M.)³.⁹², determine the maximum mass , of a star that is stable against disruption by radiation pressure.

Answer 1

`\[ (M)/(M_(\odot)) = \left( (4 \pi G M c)/(1.02 \kappa L_(\odot)) \right)^(1/3.92) \]`

This expression gives the maximum mass of a star that is stable against disruption by radiation pressure based on the mass-luminosity relation for high-mass stars and the Eddington limit.


To determine the maximum mass of a star that is stable against disruption by radiation pressure, we need to set the gravitational force equal to the radiation pressure. The Eddington limit is the point at which the radiation pressure balances the gravitational force, preventing further collapse. The Eddington limit is given by:

`\[ L_{\text{Edd}} = (4 \pi G M c)/(\kappa) \]`

where:

-
`\( L_{\text{Edd}} \)` is the Eddington luminosity,

-
`\( G \)` is the gravitational constant,

-
`\( M \)` is the mass of the star,

-
`\( c \)` is the speed of light,

-
`\( \kappa \)` is the opacity of the material in the star.

The mass-luminosity relation for high-mass stars is given by:

`\[ (L)/(L_(\odot)) = 1.02 \left( (M)/(M_(\odot)) \right)^(3.92) \]`

where:

-
`\( L_(\odot) \)` is the solar luminosity,

-
`\( M_(\odot) \)` is the solar mass.

Now, we set
`\( L \)` in the mass-luminosity relation equal to
`\( L_{\text{Edd}} \)` and solve for
`\( M \)`:

`\[ 1.02 \left( (M)/(M_(\odot)) \right)^(3.92) = (4 \pi G M c)/(\kappa L_(\odot)) \]`

Solving this equation requires specific values for the constants
`\( G \), \( c \), \( L_(\odot) \), and \( \kappa \)`. Since these values are constants, we can simplify the equation further by dividing both sides by
`\( 1.02 \)` and rearranging:

`\[ \left( (M)/(M_(\odot)) \right)^(3.92) = (4 \pi G M c)/(1.02 \kappa L_(\odot)) \]`

Now, you can take the 3.92-th root of both sides to solve for M:

`\[ (M)/(M_(\odot)) = \left( (4 \pi G M c)/(1.02 \kappa L_(\odot)) \right)^(1/3.92) \]`

This expression gives the maximum mass of a star that is stable against disruption by radiation pressure based on the mass-luminosity relation for high-mass stars and the Eddington limit.

The probable question can be: 9. [EOC-21.3] [5 points) The Eddington limit applies to stars as well as to accreting black holes, and places an upper limit on their mass. Using the mass-luminosity relation for high-mass stars: L/L = 1.02(M/M.)3.92, determine the maximum mass of a star that is stable against disruption by radiation pressure.