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When a star ignites ( ie. fusion starts ), the star maintains its form by balancing gravity's inward pressure, and radiation's outward pressure.
I get that the fusion of hydrogen atoms releases energy... fine...
How does gravity keep it together if the mass is lessening as a result of fusion( mass being converted into energy from fusion) while gravity is weakening( as mass lessens )?
Wouldn't the radiation overpower the force of gravity and tear the star apart?
The Sun's luminosity is $3.8\times 10^{26}$ W. Application of mass energy equivalence tells you it loses mass at a rate of 4.25 million tonnes per second as hydrogen turns into helium.
This is practically nothing as far as the structure of the star goes. Over its lifetime, the Sun has lost about 0.03% of its mass in this way.
Radiation pressure is a feature in stellar evolution calculations. It is almost negligible in the solar interior (at the 1% level compared with thermal pressure). However, it does become more important in more massive stars with hotter interiors and higher luminosities.
The most important fusion process in nature is the one that powers stars. In the 20th century, it was realized that the energy released from nuclear fusion reactions accounted for the longevity of the Sun and other stars as a source of heat and light. The fusion of nuclei in a star, starting from its initial hydrogen and helium abundance, provides that energy and synthesizes new nuclei as a byproduct of that fusion process. The prime energy producer in the Sun is the fusion of hydrogen to form helium, which occurs at a solar-core temperature of 14 million kelvin. The net result is the fusion of four protons into one alpha particle, with the release of two positrons, two neutrinos (which changes two of the protons into neutrons), and energy. Different reaction chains are involved, depending on the mass of the star. For stars the size of the sun or smaller, the proton-proton chain dominates. In heavier stars, the CNO cycle is more important.
Either way, the net result is 4 protons ($^1\!$H nuclei) are turned into 1 alpha particle ($^4\!$He nucleus) plus 2 positrons (e$^+$). The 2 positrons go on to annihilate 2 electrons, so altogether we have a mass change of $$ \Delta M = M_{\mathsf \alpha} - 2M_{\mathsf e} - 4M_{\mathsf P}\,. $$
Let's find out the fractional change in mass: $$ f_\Delta = \frac{\Delta M}{4M_{\mathsf P}} = \frac{M_{\mathsf \alpha} - 2M_{\mathsf e} - 4M_{\mathsf P}}{4M_{\mathsf P}}\,. $$
Now the ratio of the mass of an alpha particle to a proton is $3.9726$, or $$ M_{\mathsf \alpha} = 3.9726\times M_{\mathsf P}\,. $$
The ratio of the mass of a proton to an electron is $1836.1$, or $$ M_{\mathsf e} = \frac{M_{\mathsf P}}{1836.1} = 0.0005446\times M_{\mathsf P}\,. $$
Substituting into the $f_\Delta$ equation, $$ f_\Delta = \frac{3.9726\times M_{\mathsf P} - 0.0011\times M_{\mathsf P} - 4\times M_{\mathsf P}}{4\times M_{\mathsf P}} = \frac{-0.0285}4 = -0.007125 = -0.7125\%\,\,.$$
So obviously, Even if all of the hydrogen were converted (only a fraction actually is) the loss of mass to the star would be too negligible to matter.
A more important mass loss for large stars is that from their stellar wind, which for very large main sequence stars (types O or B) removes a sizable fraction of the very large star's mass over it's lifetime.
Some good answers, I'm going to give kind of summary, cause you touched on a few points.
Why does gravity increase in star formation
Gravitation is a product of a few forces. Mass, density and, not to be ignored, rotation speed.
It's not actually the fusion process that keeps the sun from contracting, at least, not directly. It's heat that keeps the star expanded. That's the balancing act. High temperature wants to expand, gravity wants to contract.
The fusion process is actually pretty slow, which is why stars like our sun have a main sequence of about 10 billion years, and a lot of the heat that a star starts out with is from the heat of formation. Potential energy gets converted to heat due to the coalescing and condensing of all that matter so stars start out hot, even before fusion begins.
In fact, a star in formation can be many times brighter than the star is during it's main sequence due to the high heat of formation. Here's an article that says the forming sun was 200 times brighter than it is now.
Young proto-stars, as a result of conservation of angular momentum, tend to rotate very fast and that fast rotation can create a bulge and increases ejection of matter. The formation process is pretty chaotic compared to the main sequence stage. Lots of ejected matter, much bigger solar storms, lots of lheat from formation, etc.
Once the main sequence stage is underway and rotation is slowed down, then there's more of a balance between heat and gravity mentioned above. The fusion process continues to add heat to the core of star which the star, convects or conduct heat away from the core into the outer layers and then, radiates from it's surface, but during the main sequence, in general, the core of the star gradually heats up and in most cases, the energy added from fusion isn't nearly strong enough to blow apart the star, unless the star is enormously large like over 150 or 200 solar masses, then the star doesn't really work without blowing off a bunch of matter. See: here.
I get that the fusion of hydrogen atoms releases energy... fine...
How does gravity keep it together if the mass is lessening as a result of fusion( mass being converted into energy from fusion) while gravity is weakening( as mass lessens )?
As others have said, mass loss by solar wind is a bigger factor especially for young and smaller stars, but there's a few factors at play. The short answer to this question is that the mass loss, at least by fusion, is quite very compared to the total mass of the star. Another factor, as hydrogen becomes helium, the core of the star becomes denser and greater density tends to be smaller and that increases gravity, but there are competing factors. The inner core grows denser as it becomes more hydrogen rich and the fusion tends to expand outwards on the outside of the helium core, so a star like our sun gets a denser inner core over time, but the layers around the core can grow hotter and larger, even as they lose mass.
Wouldn't the radiation overpower the force of gravity and tear the star apart?
As mentioned above, this happens if you have 150 or 200 solar masses. lower mass stars, the fusion isn't nearly powerful enough to blow the star apart. Stars and white dwarfs blow apart when they go supernova, but that's different than the main sequence fusion process.
Our sun will blow off some of it's matter when it has it's helium flash, so there are examples of what you're describing happening, but not during the main sequence for stars like our sun when material is expelled primarily by magnetic storms causing coronal mass ejections. Fusion is, generally speaking, more like a slow burn, than a big explosion when it's up against the enormous gravitational binding energy of a star.
Here's the bare bones reason for stars like our sun. The full story is much more...full.
Expansion means cooling. Cooling means less fusion. Less fusion means less energy driving expansion, meaning the outward pressure is going down. Eventually gravity is pulling inward more strongly than radiation is pushing outward. So the material collapses again. Collapsing means heating. Heating means more fusion. More fusion means more radiation pushing outward on the star. Produce enough energy, and you'll overcome gravity and the star expands.
Rinse and repeat.
The star naturally sits at an equilibrium where gravity and radiation balance each other. Deviations from this are self-correcting.
As a star runs out of hydrogen fuel, the fusion slows, causing the gravity to overpower the outward force of pressure, thus contracting. Contraction of the star causes high temperature and pressure, to the extent that it is enough to fuse helium into carbon, then the energy released is stronger than the gravity, increasing the size of the star into a red giant. The following paragraph from an article explains this:
Over its life, the outward pressure of fusion has balanced against the inward pressure of gravity. Once the fusion stops, gravity takes the lead and compresses the star smaller and tighter. Temperatures increase with the contraction, eventually reaching levels where helium is able to fuse into carbon. Depending on the mass of the star, the helium burning might be gradual or might begin with an explosive flash. The energy produced by the helium fusion causes the star to expand outward to many times its original size.
The amount of mass lost is more due to stellar wind, rather than to fusion. To answer your second question, the pressure will never overcome completely the force of gravity. When a star reaches the iron nickel stage of fusing, it'll stop, unable to go further. This causes a rapid contraction for either a star to go supernova (which is really tearing most of the star apart except for its core), or cool down to a black dwarf.
