Calculating luminosity density of the universe

[IRNB]

Homework Statement

l = Phi*L*Gamma(a+2)

b)

The Sloan Digital Sky Survey (SDSS) has recently measured the following Schechter
parameters in the r passband: a = -1.16 +/- 0.03, M*= -20.80 +/- 0.03, Phi* = (1.50 +/-0.13). Given that the sun has absolute magnitude M = 4.62 in the SDSS r band, calculate the luminosity density in this band in solar units. Estimate the error on this quantity.
Gamma(0.84) = 1.122, Gamma(0.81) = 1.153, Gamma(0.87) = 1.094.

Homework Equations

The Attempt at a Solution

I done the entire question and then realized i had made a grave error. And this is what I think it is; I took L* as being -20.80 +/- 0.03 ie M*, and I am pretty sure this is wrong. Luminosity confuses me and I'm not sure how to relate the absolute magnitude M* to the Luminosity L*. I know that L* is supposed to be in the order of 10^10. Can anyone out there help? Its kind of urgent (due in tomorrow!).

 

[IRNB]

Never mind. Problem solved

 

[nlightner]

First, let's define some terms to help us understand the problem:

- Luminosity density: This refers to the amount of light emitted per unit volume in a given region of the universe. It is typically expressed in units of solar luminosity per cubic megaparsec (M☉/Mpc^3).
- Schechter parameters: These are parameters used in the Schechter function, which is a mathematical function that describes the distribution of galaxies in a given region of the universe. The parameters include the characteristic luminosity (L*), the faint-end slope (a), and the normalization factor (Φ*).
- Absolute magnitude: This is a measure of the intrinsic brightness of an object, independent of its distance from us. It is defined as the apparent magnitude (how bright the object appears to us) that the object would have if it were located at a distance of 10 parsecs (pc) from us.
- Gamma: This is the gamma function, a mathematical function used to calculate the luminosity density in the universe.

Now, let's address the question at hand. The first thing we need to do is calculate the luminosity L* using the given absolute magnitude M* and the distance to the sun, which is approximately 1 AU (astronomical unit) or 1.5 × 10^8 km. We can use the equation L = 4πd^2F, where L is luminosity, d is distance, and F is flux.

We know that M* = 4.62 and the distance to the sun is 1 AU, so we can rearrange the equation to solve for F:

F = M*/(-2.5) = 10^(M*/(-2.5))

Substituting in the values, we get F = 10^(4.62/(-2.5)) = 0.0026. Now, we can plug this value into the equation for luminosity:

L* = 4π(1.5 × 10^8)^2(0.0026) = 5.89 × 10^26 watts

Next, we can use the given values for the Schechter parameters to calculate the luminosity density using the equation l = Φ*L*Γ(a+2). We know that a = -1.16, Φ* = 1.50, and Γ(0.84) = 1.