Has the loss of mass by the Sun over the last 4 billion years been enough to affect planetary orbits?

The luminosity of the Sun is 200 trillion trillion watts or 2 x 10^33 ergs per second. From Einstein's famous equation, E = mc^2, and using c = 3 x 10^10 centimeters/sec, the Sun's luminosity is equal to a loss of mass from the fusion cycle of about 2 x 10^12 grams/second. Over one year this is 7 x 10^19 grams, and over the entire life of the Sun to date is about 3.1 x 10^29 grams. The mass of the Sun is 4 x 10^33 grams so this loss equals 0.008 percent of its current mass. The mass of Jupiter is about 0.1 percent of the Sun's current mass, so over the Sun's entire lifetime to date, it has lost barely 0.08 percent of Jupiter's mass, or about the mass of the Earth.

We can estimate how much this mass loss would have changed the orbit of the Earth by approximating the orbital dynamics as the balance between kinetic and gravitational potential energy or 1/2 mV^2 = GMm/R where m = mass of Earth, and M = mass of Sun. We see that a reduction in the Sun's mass by a factor of of 0.00008 causes an increase in the Earth-Sun distance if the kinetic energy of the Earth is held constant. This means that over the last 4.5 billion years, we can estimate that the Earth's orbit has increased by about 0.00008 x 93 million miles or about 7,000 miles; about the Earth's own diameter!

The Sun also produces a 'solar wind' of particles at a rate of about 10^-14 solar masses per year. In 4 billion years this amounts to about 0.001 percent of the Sun's mass, which to the level of our approximations is a factor of 8 times smaller that the mass loss from converting some of its mass into light. However, both the solar luminosity and solar wind have not been constant over 4 billion years, with the sun having been fainter long ago, and its wind having been much stronger when it was first born.

The overall effects of these mass loss rates can be significant when dynamicists try to predict the long-term orbits of planets. We know that small changes in any physical parameter in these 'non-linear' mathematical theories can produce substantial changes in the locations of planets in their orbits. It would not surprise me if the sun loosing 1 Earth mass over 4 billion years might also have a significant effect in predictions of where planets are in the distant future.

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