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A Luminosity

Luminosity of Star:

The luminosity is the total energy given off by a blackbody per second (i.e. the power). The product from Stefan-Boltzmann Law (\(\sigma T^4\)) is only the energy per second per unit surface area. To obtain the luminosity, we must multiply it by the surface area. If the star is a sphere with radius R, the surface area is (\(4\pi R^2\) ), so the luminosity is:

\[L=(\sigma T^4)(4\pi R^2)\]

Stars become larger with increasing surface temperature. This is why the luminosity of stars increases with increasing surface temperature at a rate greater than (\(T^4\)).

Derivation of the Stefan-Boltzmann law:

This involves a more rigorous treatment of black body radiation and the principles of statistical mechanics.

The Stefan-Boltzmann law is derived from the Planck radiation formula, and the key steps involve:

  1. Density of States: First, we consider the density of states, which is a measure of the number of available energy states for photons in the black body. It is usually expressed as a function of frequency, g(ν)dν, which represents the number of states per unit frequency interval.
  2. Energy Distribution: Next, we consider the energy distribution of photons. According to Bose-Einstein statistics, which are appropriate for photons, the average number of photons in a given state follows:

\[n(\nu)=\frac 1 {e^{h\nu /kT}-1}\]

Where:

  1. Energy Density: To find the energy density at a given frequency, we need to consider both the energy per photon and the number of photons at that frequency. The energy per photon is E=hν. So, the energy density u(ν) is given by:

\[u(\nu)=\frac EV g(\nu )n(\nu)\]

Where:

  1. Integrate Over All Frequencies: To find the total energy density u (integrated over all frequencies), you need to integrate u(ν) over all possible frequencies. This integration is typically carried out from 0 to infinity.

\[u=\int _0 ^\inf \frac EV g(\nu )n(\nu)d\nu\]

  1. Stefan-Boltzmann Law: The Stefan-Boltzmann law relates the total power radiated per unit area (P) to the total energy density (u) and is given by:

\[P=\sigma T^4\]

Where:

The Planck radiation formula is given by:

\[u(\nu)=\frac {8\pi \nu ^2}{c^3}\frac {h\nu}{e^{h\nu /kT}-1}\]

Where:

Now, we want to find the total energy density (u) by integrating this formula over all frequencies.

  1. Start by defining the energy density �u integrated over all frequencies:

\[u=\int_0 ^\inf u(\nu)d\nu\]

  1. Plug in the expression for u(ν) from the Planck radiation formula:

\[u=\int _0 ^\inf \frac {8\pi \nu ^2}{c^3}\frac {h\nu}{e^{h\nu /kT}-1}d\nu\]

  1. Now, we need to perform this integral. To simplify it, we can make a change of variable. Let \(x=h\nu/kT\)​, which gives \(\nu =kTx/h\) and \(d\nu =kTdx/h\):

\[u=\int _0 ^\inf \frac {8\pi \frac {k^2T^2x^2}{h^2}}{c^3}\frac {h \frac {kTx}{h}}{e^x-1} \frac{kT}{h} dx\]

\[u=\frac {8\pi k^4}{c^3 h^3}T^4\int _0 ^\inf \frac {x^3}{e^x-1} dx\]

Now, the integral is known as a special function called the polylogarithm function, denoted as \(Li_3(z)\), where \(z=e^{-x}\) . So \(x=-ln(z)\)and \(dx=-\frac {dz}{z}\)and the integral can be written as:

\[u=\frac {8\pi k^4}{c^3 h^3}T^4\int _1 ^0 \frac {-{ln(z)}^3}{\frac 1z-1} (-\frac {dz}{z})\]

\[u=\frac {8\pi k^4}{c^3 h^3}T^4\int _0 ^1 \frac {{ln(z)}^3}{z-1} dz\]

Solar Luminosity:

The surface temperature of the Sun is about 5800 K and its radius is 7 × 10^5 km (7 × 10^10 cm). So the luminosity of the Sun:

\[L=4\pi (7\times 10^{10}cm)^2[5.7\times10^{-5}\ erg/(cm^2K^4s)]\times (5.8\times 10^3 K)^4\]

\[L=4 \times 10^{33}\ erg/s\]

This quantity is called the solar luminosity, and L, serves as a convenient unit for expressing the luminosities of other stars.