The bigger the star's radius, the greater its surface area which emits the light. The bigger the temperature, the more luminous is the light the star is emitting.
If the radius is larger, the surface will also be larger. As a functional dependency, you only need one - the radius, or the surface - whatever.
As temperature decreases, luminosity will also decrease As radius increases (and with it surface area, but radius is a much easier to work with if you're trying to compare stars so we usually say radius) luminosity will also increase. If both are happening at the same time, it is possible that the luminosity of the star will remain more or less constant. Often one change will dominate the other, such as when a star goes through the red giant phase when the increase in radius has a far greater effect than the drop in temperature, and the star becomes more luminous.
Both the absorption and the luminosity of a blackbody in equilibrium increase in magnitude with increasing temperature, and the spectral distribution of the luminosity increases in frequency (decreases in wavelength).
As gravity collapses the cloud to form a protostar, the temperature and luminosity both increase. The increase in temperature is due to the compression of material, causing the protostar to heat up as energy is released. The increase in luminosity is a result of the protostar radiating this energy.
If a star has a large luminosity and a low surface temperature, it must have a large surface area to compensate for the low temperature and still emit a high amount of energy. This would make the star a red supergiant, a type of star that is both luminous and cool at the same time.
Sirius A and Procyon A are two stars that have similar luminosity and surface temperature. They are both main-sequence stars and are relatively close to each other in terms of these characteristics.
No. Main sequence stars vary greatly in both temperature and luminosity. The least massive stars, red dwarfs, can have temperatures as low as 2,300 Kelvin and luminosity as low as 0.015% that of the sun. The most massive stars, which are blue in color can have temperatures as high as 50,000 Kelvin and may be hundreds of thousands times more luminous than the sun.
In that case, both the star's diameter and its luminosity greatly increase.
The scatter plot of the relationship between a star's temperature and luminosity is represented by the Hertzsprung-Russell diagram. In a standard H-R diagram the horizontal axis shows the [surface] temperature, increasing from right to left, while the vertical axis shows luminosity increasing from bottom to top. When both axis are on a logarithmic scale, the main sequence stars from a diagonal belt stretching from top right (very hot and very luminous) to bottom left (not so hot and not so luminous).
The temperature in both cases will depend on the exact details of the setup.
Yes, both volume and mass can depend on temperature. Volume can change with temperature due to thermal expansion or contraction, while the mass of a substance remains constant regardless of temperature. However, changes in temperature can affect the density of a substance, which is the mass per unit volume.
In both cases, the radius can be any positive number.