|MadSci Network: Astronomy|
The short answer to your question is yes, the Sun is losing mass over time. However, the amount of mass it loses over time is insignificant.
As you pointed out, the Sun radiates away energy that is produced by a nuclear fusion reaction converting hydrogen nuclei into helium nuclei. In each reaction 4 hydrogen nuclei (4 protons) are fused into a single helium nucleus (2 protons and 2 neutrons). The helium nucleus does have less mass than the 4 protons, and the extra mass is converted into energy. The fraction of mass converted into energy in this reaction is 0.68%. So it is a very small amount of mass lost. Also, the Sun cannot use all of it mass for fusion. Only the hydrogen in the core of the Sun that is hot and dense enough can undergo this reaction. Over the lifetime of the Sun only 10% of the Sun's mass will undergo fusion. So that means that over the 10 billion year lifetime of the Sun only 0.068% of the Sun's mass will be lost to fusion.
Now, there is another important source of mass loss for the Sun. The Sun blows a constant wind of gas off its surface into interplanetary and interstellar space. The Sun loses about 2x1017 kg per year due to this wind (note: here I have used scientific notation to represent a very large number. What I have written is equivalent to 2 with 17 zeros following it, a number so large it has no name). For reference the Sun has a mass of 2x1030 kg, and Earth has a mass of 6x1024 kg.
The Sun's mass loss rate due to fusion is 1.3x1017 kg per year. So the total mass loss rate is about 3.3x1017 kg per year. Over 10 billion years (1010 years) this would add up to 3.3x1027 kg lost by the Sun. This is only a loss of 0.17% of the Sun's total mass. Not a significant fraction at all. (note: this mass lost is equivalent to 550 Earths, but it is still an insignificant amount to the Sun).
As to your point about the space dust that falls to Earth possibly adding to the mass of the Sun, this is probably not a significant effect. Most of this dust is actually blown back from the Sun's surface by the solar wind. There have been comets that have been observed to crash into the Sun presumably adding their mass to the Sun. Much of these comets' mass would also have been blown back by the solar wind. This is a rare event in any case. Still we can make a simple estimate that one comet crashes into the Sun per year adding to its mass. This would come out to about 5x1014 kg per year. That's a pretty liberal estimate really and its still nearly one thousand times smaller than the total mass lost per year. So the net effect would still be mass loss.
Either way, the Sun's mass (and the Earth's too) is not changing by any significant amount, nor will it until it dies as a Red Giant in some 5 billion years (we won't be around then). So don't worry about it.
Thanks for your question,
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