Using Eddington Luminosity to calculate mass

In summary: I am not sure!In summary, the nuclear reactions in a star’s core are sensitive to pressure and temperature, resulting in higher luminosities for high-mass stars. The luminosity of massive stars follows a scaling relationship with mass, as shown by the equation Lstar = (34.2)*M^2.4, where M and L have units of MSun and LSun, respectively. The Eddington limit, which represents the maximum luminosity an object can achieve while maintaining a balance between outward energy and inward gravity, is given by the equation LEdd = (3.2*10^4)*M, also in units of MSun and LSun. To calculate the maximum mass of a star in solar units
  • #1
Ellie Snyder
8
1

Homework Statement


The nuclear reactions in a star’s core are very sensitive to pressure and temperature, so high mass stars have much higher luminosities. The luminosities of massive stars have been observed to obey the following scaling relationship with mass (M):

Lstar = (34.2)*M^2.4, where M, L have units of MSun, LSun.

The Eddington limit is the maximum luminosity an object (such as a star) can achieve and still retain a balance between the outward force of energy from the center and inward pull of gravity. The following formula is for the Eddington Limit (LEdd), i.e., the luminosity which stops the inward pull of gravity:

LEdd = (3.2*10^4)*M, where M, L have units of MSun, LSun.

a. Using these two equations, calculate the maximum mass of a star in solar units.

b. A bright quasar has a luminosity of about 10^13 LSun. The source of its power is a supermassive black hole that attracts surrounding gas into a hot (~10^6 K), compact accretion disk which radiates light. If the quasar is to continue attracting gas into its central black hole, what is its minimum mass?

Homework Equations


Lstar = (34.2)*M^2.4
LEdd = (3.2*10^4)*M

The Attempt at a Solution


a. Solved for M in the Lstar eqn to get (Lstar/34.2)^1/2.4 and plugged that into the LEdd eqn. Got 7344.7Lstar^1/2.4, and plugged that in for LEdd to get an M of 0.2295Lstar^1/2.4, which is obviously preposterous. I guess I'm not sure how to relate the equations to solve for M.

b. Waiting to attempt this one until I have a better understanding of a.
 
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  • #2
Ellie Snyder said:

Homework Statement


The nuclear reactions in a star’s core are very sensitive to pressure and temperature, so high mass stars have much higher luminosities. The luminosities of massive stars have been observed to obey the following scaling relationship with mass (M):

Lstar = (34.2)*M^2.4, where M, L have units of MSun, LSun.

The Eddington limit is the maximum luminosity an object (such as a star) can achieve and still retain a balance between the outward force of energy from the center and inward pull of gravity. The following formula is for the Eddington Limit (LEdd), i.e., the luminosity which stops the inward pull of gravity:

LEdd = (3.2*10^4)*M, where M, L have units of MSun, LSun.

a. Using these two equations, calculate the maximum mass of a star in solar units.

b. A bright quasar has a luminosity of about 10^13 LSun. The source of its power is a supermassive black hole that attracts surrounding gas into a hot (~10^6 K), compact accretion disk which radiates light. If the quasar is to continue attracting gas into its central black hole, what is its minimum mass?

Homework Equations


Lstar = (34.2)*M^2.4
LEdd = (3.2*10^4)*M

I think the correct equation for the Eddington Luminosity Limit is

LEdd = (3.2×104) ⋅ (Mstar / Msun) ⋅ Lsun

https://en.wikipedia.org/wiki/Eddington_luminosity
 
  • #3
SteamKing said:
I think the correct equation for the Eddington Luminosity Limit is

LEdd = (3.2×104) ⋅ (Mstar / Msun) ⋅ Lsun

https://en.wikipedia.org/wiki/Eddington_luminosity
Hmm. I believe you but since I was given that formula in the question itself, I think I should use it to answer the question, even if it is off base.
 
  • #4
Ellie Snyder said:
Hmm. I believe you but since I was given that formula in the question itself, I think I should use it to answer the question, even if it is off base.
Well, the factor which was missing from your original equation was the luminosity of the sun. Perhaps it got lost when you were drafting your post.
 
  • #5
SteamKing said:
I think the correct equation for the Eddington Luminosity Limit is

LEdd = (3.2×104) ⋅ (Mstar / Msun) ⋅ Lsun

https://en.wikipedia.org/wiki/Eddington_luminosity
Ellie specified that M and L were in solar units (hence Msun=1 and Lsun=1) so the expression is the same.

@Ellie Snyder
You solved for M in terms of L_star, but L_star is unknown, so that isn't a useful thing to do.

I think you've realized that the maximum mass occurs when L_star = L_Edd, right? So then algebraically you can just equate those two expressions:
##L_{star}=L_{Edd}\Rightarrow 34.2M^{2.4}=32000M##
So that way the equation involves only M and not L.
 
  • #6
Nathanael said:
Ellie specified that M and L were in solar units (hence Msun=1 and Lsun=1) so the expression is the same.

@Ellie Snyder
You solved for M in terms of L_star, but L_star is unknown, so that isn't a useful thing to do.

I think you've realized that the maximum mass occurs when L_star = L_Edd, right? So then algebraically you can just equate those two expressions:
##L_{star}=L_{Edd}\Rightarrow 34.2M^{2.4}=32000M##
So that way the equation involves only M and not L.
Oh duh, ok great, so I'm getting an M of about 132.50 M_sun
 
  • #7
Nathanael said:
Ellie specified that M and L were in solar units (hence Msun=1 and Lsun=1) so the expression is the same.

@Ellie Snyder
You solved for M in terms of L_star, but L_star is unknown, so that isn't a useful thing to do.

I think you've realized that the maximum mass occurs when L_star = L_Edd, right? So then algebraically you can just equate those two expressions:
##L_{star}=L_{Edd}\Rightarrow 34.2M^{2.4}=32000M##
So that way the equation involves only M and not L.
So for part b, to find that minimum mass, would I insert the quasar's 10^13 L_sun luminosity into the first equation and solve for M?
 
  • #8
Ellie Snyder said:
So for part b, to find that minimum mass, would I insert the quasar's 10^13 L_sun luminosity into the first equation and solve for M?
I think so, but to be honest, I don't entirely understand the physics behind that situation; it seems like some subtleties are being neglected.
 
  • #9
I have the same questions, just put into a different scenario... I ended up taking the 1.4 root of 3.2x10^4 over 34.2 for an answer of 266, 000kg. Because i rounded to 133 Msun which makes 133x2x10^3kg...Is this completely off? My professor said I was right on... but now I am wondering if I should just leave it as 133Msun...?
 
  • #10
WHY do I over think everything? Its the same damn thing... nevermind me I'm just arguing with myself...
As usual...
 

Related to Using Eddington Luminosity to calculate mass

1. How does Eddington Luminosity help in calculating mass?

Eddington Luminosity is a theoretical limit of the maximum luminosity a star can have before it becomes unstable and starts losing mass. By measuring the luminosity of a star and comparing it to the Eddington Luminosity, we can estimate the mass of the star.

2. What is the formula for calculating mass using Eddington Luminosity?

The formula for calculating mass using Eddington Luminosity is: M = (4πGM)/(κc), where M is the mass of the star, G is the gravitational constant, κ is the opacity of the star's gas, and c is the speed of light.

3. Can Eddington Luminosity be used to calculate the mass of any star?

No, Eddington Luminosity can only be used to calculate the mass of stars that are in a state of hydrostatic equilibrium, meaning their outward radiation pressure is balanced by their inward gravitational force.

4. Are there any limitations to using Eddington Luminosity to calculate mass?

Yes, there are a few limitations to using Eddington Luminosity. The formula assumes that the star is spherical and has a uniform density, which may not always be the case. Additionally, the opacity of the star's gas can vary and affect the accuracy of the calculation.

5. How accurate is using Eddington Luminosity to calculate mass?

The accuracy of using Eddington Luminosity to calculate mass depends on the assumptions made and the limitations of the formula. It can provide a rough estimate of the mass, but for more precise measurements, other methods may need to be used.

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