Hertzsprung – Russell Diagram A plot of the luminosity as a function of the surface temperature for different radii stars.

Show all the steps, including the non-dimensionalization process, to demonstrate that a plot of the Log (L) vs Log (T) is linear for stars of equal radius. Begin with Stefan’s Law: Hertzsprung – Russell Diagram L A = T4T4 Where L is the luminosity in Watts A is the surface area F is the Stefan-Boltzmann constant = 5.67 x W/m 2 – K 4 T is the surface temperature in Kelvin

Hertzsprung – Russell Diagram L A = T4T4 L=4  R 2  T 4 Show all the steps, including the non-dimensionalization process, to demonstrate that a plot of the Log (L) vs Log (T) is linear for stars of equal radius. Re-arrange Stefan’s Law: It would seem that we could simply take the Log of both sides, but the Log of a number that has a dimension is not defined. Example: How does Log (4 Joules) compare with Log (4 kg)? The question has no meaning

Hertzsprung – Russell Diagram Show all the steps, including the non-dimensionalization process, to demonstrate that a plot of the Log (L) vs Log (T) is linear for stars of equal radius. The process begins by making the right and left hand sides of Stefan’s Law dimensionless. This is done by multiplying and dividing by some appropriately defined “standard.” The standards used in a H-R diagram are the luminosity, radius and temperature of the sun, symbolized as LL RR TT Where  is the symbol for the sun

Hertzsprung – Russell Diagram L =4   LL LL R2R2 T4T4 R2R2 R2R2 T4T4 T4T4 ()))(( L = LL LL R2R2 T4T4 R2R2 R2R2 T4T4 T4T4 ()))(( = LL R2R2 T4T4 But: Show all the steps, including the non-dimensionalization process, to demonstrate that a plot of the Log (L) vs Log (T) is linear for stars of equal radius. The left side is multiplied and divided by luminosity of the sun, and the right side is multiplied and divided by the radius of the sun squared and the temperature of the sun to the fourth power.

Hertzsprung – Russell Diagram L = LL R2R2 T4T4 R2R2 T4T4 ())( T4T4 If luminosity, temperature and radius are given as the fraction of the luminosity and radius of the sun =L fractional T fractional 4 R fractional 2 Where L fractional,T fractional and R fractional are the fraction of the suns luminosity, temperature and radius. Show all the steps, including the non-dimensionalization process, to demonstrate that a plot of the Log (L) vs Log (T) is linear for stars of equal radius. Therefore:

AND, using logarithms: Hertzsprung – Russell Diagram =Log (L fractional )T fractional 4 R fractional 2 Log () =Log (L fractional )4 Log T Fractional R fractional 2 + Log () y = mx + b Note: The slope is +4, which would seem to produce a line that runs from the lower left to the upper right. The line is reversed from the norm, however, because the units of the x axis are reversed.