how can white dwarf stars be both hot and dim?

White dwarf stars can be both hot and dim because of their small size and the fact that they no longer undergo nuclear fusion.

A white dwarf is the remnant core of a medium-sized star, like our Sun, after it has exhausted its nuclear fuel and shed its outer layers as a planetary nebula. When the core is left to cool, it becomes a white dwarf.

Despite their high surface temperatures, which can exceed tens of thousands of degrees Kelvin at formation, white dwarfs are relatively dim. The primary reason for this apparent contradiction—high temperature but low luminosity—is their small radius, which is typically about the same size as Earth's. Since luminosity (the total energy output of a star) is a product of both surface area and temperature, the small surface area of a white dwarf limits its brightness.

So, even though its surface is very hot, because the overall size of the star is small, it emits much less total light than a larger star at the same temperature. This is why we perceive white dwarfs as relatively dim objects in the sky, even though they have high surface temperatures.

Extra: The concept of brightness in astronomy can be split into two terms: luminosity and apparent magnitude. Luminosity is the intrinsic brightness of a celestial object, referring to the total amount of energy emitted by the object each second. It's equivalent to the power output of the star. Apparent magnitude, on the other hand, measures how bright the object appears from Earth, which can be affected by distance and other intervening factors.

In the case of white dwarf stars, their intrinsic luminosity is low because of their small size. This intrinsic property does not change regardless of how far away the white dwarf is from us. However, the apparent magnitude would vary with distance – the greater the distance, the dimmer the white dwarf appears.

When studying white dwarf stars, astronomers also take into account a concept known as the Stefan–Boltzmann Law, which states that the energy radiated per unit surface area of a black body (an idealized physical body that absorbs all incident electromagnetic radiation) is proportional to the fourth power of its temperature. This principle helps explain why, despite their high temperatures, white dwarfs do not have high luminosity – the amount of surface area available to radiate energy is relatively small compared to larger stars.