About initial mass function and mean mass in stellar cluster

[Cedric Chia]

Homework Statement

Assuming a Salpeter IMF with upper and lower mass limits of 0.1 and 20 M⊙ respectively, calculate:

(i) the mass point at which half the mass formed in a stellar cluster lies in more massive systems and half in less massive systems.

ii) the mass point at which half the luminosity of the star cluster lies in more massive systems and half in less massive systems.

Homework Equations

to find number of stars in the range x and y: f_N = ∫_x^y M ^-7/3 dM / ∫_0.1²⁰ M ^-7/3 dM
to find mass occupied by the starts in the range x and y: f_M = ∫_x^y M ^-7/3 dM / ∫_0.1²⁰ M ^-7/3 dM

The Attempt at a Solution

Question(i) :
using the second equation,
f_M = ∫_x^y M ^-7/3 dM / ∫_0.1²⁰ M ^-7/3 dM
1/2 = ∫_x²⁰ M ^-7/3 dM / ∫_0.1²⁰ M ^-7/3 dM
solving for x and we get,
x = 0.5 M⊙

 

[mfb]

Your two relevant equations look the same but the exponent should be different.

 

[Cedric Chia]

Your two relevant equations look the same but the exponent should be different.

yess I'm really sorry I've forgot to put another M in the second equation!

correction:
to find mass occupied by the starts in the range x and y: f_M = ∫_x^y M^-7/3 M dM / ∫_0.1²⁰ M^-7/3 M dM

 

[mfb]

Looks good.