# Mass Lost by the Sun by emitting radiation

## Main Question or Discussion Point

Hi,

I recently did a homework problem for my introductory Quantum Mechanics course and in the problem we were given the sun's temperature in Kelvin as well as the diameter of the sun and instructed to calculate the rest mass lost per second to radiation by the sun. I was able to solve the problem after a bit of work but a question stuck out in my head while I was doing it. If we were to calculate the amount of mass lost by the sun caused by it's emitting radiation (light) then how exactly does or could the sun lose mass solely through the emission of mass-less particles? Essentially I don't quite understand how you can give off mass-less particles and lose mass because of it. Could somebody help clarify a bit for me?

I have a feeling that E=mc$^{2}$ would have to do with it because the equation explains the mass-energy equivalence. But I'm still not sure where along the line that the mass get's lost. Is it essentially that some amount of mass inside of the sun is converted to energy and that energy being radiant energy from electromagnetic radiation? Anyway, any help in clarification would be much appreciated.

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PAllen
2019 Award
There are two aspects to understanding this. The first is one you hint at: energy is emitted, so the total mass of the sun in its COM frame is reduced by E/c^2. For a more complete understanding, you then want to look at the invariant mass of the emitted photons. For this you need to know that (in SR) the total 4-momentum of a system of particles is just the vector sum of the 4-momenta of each particle. The 4-momentum of a photon is (E/c,p), where p=E/c; its norm (invariant mass) per the Minkowski metric is (E/c)^2 - p^2 = 0, as expected. Note that p is directional quantity, and can be positive or negative along any axis. However, for the system of all photons emitted by the sun, adding their 4 momenta produces a vector sum with p=0 (because the directions of all the particles cancels, assuming isotropic emission), while the energies add. The result is the the invariant mass of the system of photons exactly equals the invariant mass lost by the sun.

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Bandersnatch
I get the feeling that PAllen's explanation will go over your head.

For a less rigorous look at the problem, consider the process of fusion. Two light nuclei are smashed against each other and combine to produce a heavier nucleus. The resultant nucleus is lighter than the parent nuclei taken together, and the difference in mass is released as EM radiation and/or neutrinos, electrons and their antiparticles(cf. http://en.wikipedia.org/wiki/Binding_energy), according to E=mc^2.

Note that this works for daughter nuclei up to the neighbourhood of nickel and iron, above which the daughter nucleus is heavier than the two parent ones, meaning that it takes energy to bind it, and energy is released when the hevy nucleus splits(which is fission).

In stars, most of the energy during their lifetime is produced by the fusion of hydrogen into helium(cf. http://en.wikipedia.org/wiki/Nuclear_fusion#Astrophysical_reaction_chains). It is the most energy efficient process, as the mass difference between four protons(hydrogen nuclei) and a single helium nucleus is the greatest. Yet the mass difference amounts to merely 0.7 percent of the component protons, meaning that if all of the star's hydrogen were to fuse(which it never does), it would only lose that much of its initial mass.

SteamKing
Staff Emeritus
Homework Helper
To further complicate matters, not only is the sun losing mass due to radiation, but the high temperature in the sun's corona accelerates some of the gas surrounding the sun away from the star and creates what is known as the solar wind. To be sure, the mass loss due to radiation is greater than the mass loss to create the solar wind.

http://en.wikipedia.org/wiki/Solar_wind

mfb
Mentor
Is it essentially that some amount of mass inside of the sun is converted to energy and that energy being radiant energy from electromagnetic radiation
Exactly. In the core, hydrogen (protons) fuses to helium, and those helium nuclei are a bit lighter than the protons that formed them. This energy is transported to the surface and radiated away. As a result, the total (rest) energy of the sun (and therefore its mass) reduces.

PAllen
2019 Award
I wanted to directly address the OP concern that if the sun loses mass, and photons are massless, where does the mass go? The simple part of the answer is the photons have energy even thought they don't have mass. But more can be said: the invariant mass of the system of emitted photons exactly matches the mass lost by the sun; that a gas of massless particles can have nonzero invariant mass.

Thank you all for your replies. PAllen's explanation did go a bit over my head but I appreciate PAllen's act of putting it here. I may not quite comprehend it that well yet but with time I'm sure I will. And thank you Bandersnatch, SteamKing, and mfb for breaking it down for me. That helped.

Interesting. I kinda figured that it would have something to do with the process of fusion that takes place in the sun. But something else I was kinda curious about, could the problem I described above apply to say a heating coil heated up to a sufficient temperature at which it would put out light? That is could it also be said that a glowing red heating coil is also losing mass due to it's emission of radiation?

PAllen
2019 Award
Thank you all for your replies. PAllen's explanation did go a bit over my head but I appreciate PAllen's act of putting it here. I may not quite comprehend it that well yet but with time I'm sure I will. And thank you Bandersnatch, SteamKing, and mfb for breaking it down for me. That helped.

Interesting. I kinda figured that it would have something to do with the process of fusion that takes place in the sun. But something else I was kinda curious about, could the problem I described above apply to say a heating coil heated up to a sufficient temperature at which it would put out light? That is could it also be said that a glowing red heating coil is also losing mass due to it's emission of radiation?
Absolutely, the light is carrying away energy, the system of photons has mass (even though each does not). However, in this case, the coil is not likely loosing mass because electrical energy is being supplied to heat the coil. The coil is acting as an energy converter. However, if the coil is, for example, powered by battery, the battery is losing mass.

I saw your question as generic and deliberately did not bring fusion into it. Any time a system losing net energy (the coil is not - it is pass through), it is losing mass. Further, the invariant mass of whatever is emitted, in total, will match the mass lost by the energy source.

(I also thought I was pitching it close to the right level because you said you were in a quantum mechanics course. I don't know order of courses where you are, but I had been presented enough SR in undergrad classical mechanics and EM to have followed what I wrote, before any course in QM. Obviously I guessed wrong.)

Very cool. I had no idea that say in the instance of the battery and the heating coil that the battery would be losing mass. Out of curiosity if you were to talk about the example of the battery and the heating coil if you were to recharge the battery, after it having been used for some amount of time, would it then gain mass?

And don't worry about it. I should have been more clear. It's an introductory quantum mechanics course in the undergrad level. The proper name of the course is Modern Physics I. This course is the first course in which I've ever really dealt with any of this on anything more than a superficial level so I'm still fairly new to it.

Nugatory
Mentor
if you were to recharge the battery, after it having been used for some amount of time, would it then gain mass?
Yes.

You're talking about a few nanograms for a lead-acid auto battery that weighs several tens of kilograms, so the effect is very hard to measure (I doubt that anyone has done the experiment for that particular configuration), but it's there.

PAllen