Question

How can I see that the the luminosity of the sun is 3.828×1026 W by looking at the HR diagram?

Answer 1

HR Diagram : It is a plot of Luminosity vs Color(or spectral type or temperature).The majority of stars fall along the ‘Main Sequence’ (fig 1) .

The HR diagram becomes a calculating tool when one realizes that temperature, luminosity and size (radius) are all related by Stefan-Boltzmann's law. The standard Stefan-Boltzmann law is for point sources, an idealized case. Real objects have size, which means surface area. Large objects cool faster (emit energy faster) than small objects, so there must be some correlation with radius. For stars, the luminosity, temperature and radius are linked by an expansion of Stefan-Boltzmann's law that says:

L = 4?R2?T4
This equation can be expressing in terms of solar units such that:

L/Lo = (R/Ro)2(T/To)4
where Lo, Ro and To are the luminosity, radius and surface temperature of the Sun.

Knowing from laboratory measurements that Stefan-Boltzmann's constant is 5.67x10-8 allows one to calculate the luminosity of a star in units of watts (like a light bulb) if we know the radius of the star in meters and the temperature in kelvins. For example, the Sun is 6.96x108 meters in radius and has a surface temperature of 5780K. Therefore, its luminosity is 3.84x1026 watts.