An overview of how main sequence stars work

A review of main sequence (MS) stars and how they work, although much of this applies to stars in general:

1. Stars shine because they're hot and dense: emit a thermal (blackbody) radiation spectrum modified by absorption lines; this absorption line spectrum is produced at its surface (called a photosphere).

the result of the gravitational collapse of a gas cloud
pressure - gravity balance

given their mass relative to their size, this means stars must be and remain hot
sets up structure within star: pressure (P), temperature (T), number density of particles (n) - in order for the star to support itself against its own weight, all three of these must reach their highest values within the star's core, and then diminish in value toward the star's surface. Recall that gas pressure P is proportional to the product of the number density of particles n and temperature T.

2. If core T exceeds 10⁷ K: 4 Hydrogen nuclei fuse into 1 Helium nucleus + energy (gamma-rays) + 2 neutrinos at a rate sufficient to balance off the loss of energy in the form of light at the star's surface (i.e., it's luminosity). In our Sun, the central temperature is at present about 15.7 million K, and 613.8 million metric tons of H are fused into 609.5 million metric tons of He each second. By the way, what happens to the other 4.3 million tons of matter processed per second?

energy generated by the star's core is transported through it, to & through the star's envelope (the portion of the star outside of its core), and then to the surface by radiation (photons) or by convection. Photons stream out of the star's photosphere into dark, cold space - that which we call star light.
energy balance & energy transport: energy lost at surface in the form of light (luminosity L_*) is replaced precisely by energy released via fusion (L_fusion) within star's core, i.e., L_*= L_fusion. In this case, the star need not contract to remain hot enough to generate sufficient pressure to avoid collapsing under its own weight.
pressure-temperature thermostat regulates fusion within core (keeps star from exploding like a thermonuclear bomb), & together with energy transport regulate the energy balance of star

3. so...higher mass main sequence stars:

require higher T at every layer within for sufficient pressure support against gravity

T at the star's surface (T_s) dictates star's spectrum (thermal radiation spectrum modifed by absorption lines)
T within core sets energy released by fusion - higher central T means greater L_fusion, thus balancing a more massive star's greater surface luminosity L_*

are larger in radius R for the following reasons. If all MS stars had the same density (i.e., the ratio mass/volume which scales as ~ M/R³), then the size of the star would be larger for a larger mass, scaling like R ~ M^1/3. However, a star that is supported against its weight by pressure that depends on temperature (such as gas pressure) cannot have both more mass and an equal or higher density. Therefore, higher mass MS stars must be less dense than lower mass ones. In fact the sizes of main sequence stars scale more rapidly with mass, as R ~ M^0.6 (with a power on mass of 0.6), than do masses of equal density (with a power on mass of 1/3). During a star's formative stage as a gravitationally contracting protostar of a given mass M, it contracts gravitationally while balancing pressure and gravity, to a size R, so that its central core temperature becomes great enough to make L_fusion = L_*, in converting hydrogen into helium in its core.
Given that L_* is proportional to R²x T_s⁴, the hotter, larger, more massive MS stars are therefore much more luminous than MS stars of lower mass. In fact the luminosities of these stars scale with their mass as L ~ M^3.5. This greater surface luminosity is precisely balanced by their greater fusion energy generation rate, L_fusion, meaning that MS stars are in a state of energy balance.
have, therefore, disproportionately rapid consumption of fuel supply (H within the central core) to produce sufficient energy to keep itself hot enough to remain in pressure-gravity balance, and so have much shorter life times
All of these characteristics are in contrast to the situation for lower mass main sequence stars.

4. If pressure - gravity balance is established with central temperatures significantly less than 10⁷ K, the full set of fusion reactions converting hydrogen into helium + energy cannot occur. The object never achieves energy balance (it radiates energy away in the form of light from its surface, but has no way to replace that lost energy), and thus this object never becomes a full-fledged star, and is instead called a brown dwarf (a failed or `still-born' star). This occurs for objects² whose masses lie below 0.08 M_sun. There are now thousands of "brown dwarfs" known, and they are suspected to be nearly as numerous as all normal stars in the Milky Way Galaxy combined.

Remember, you can compare the structures of only those stars that have the same energy source in their cores. For main sequence stars, this is the fusion of hydrogen into helium in their central most regions. Can you see how the many known observed facts of main sequence stars are logically and causally linked together by the laws of nature?

¹Strictly speaking, this is true when averaged over the characteristic time required to transport the energy through the star. If for any reason the star isn't in energy balance, you can bet that it is adjusting its structure (expanding/cooling or contracting/heating) until it is.
²By this time these objects' densities have become so high that a weird quantum mechanical pressure known as "electron degeneracy pressure" becomes important. Since this kind of pressure does not depend on temperature, it can balance against the force of gravity even though the object continues to radiate energy away as light from its surface. As a consequence the object does not contract, and without contraction no more gravitational potential energy can be released to raise the temperature, and so the temperature slowly falls over the eons of time.

Kirk T. Korista
Professor of Astronomy
Department of Physics
Western Michigan University
email: kirk.korista@wmich.edu