Misunderstanding of how the Sun losing mass makes the Earth orbit increases in size

Question

The radius of the Earth orbit increases by about 15 cm per year. One of the main argument found online to explain this is the fact that the gravitational pull of the Sun on the Earth diminishes over time as the Sun looses mass trough nuclear fusion.

Now I would assume that the total energy of the Earth-Sun system remains unchanged as the change in the mass of the Sun is an intrinsic process. Since the Sun is much more massive than the Earth, I think we can think of the energy of the system as just Earth’s energy, in which case whatever happens to the Sun shouldn’t affect Earth energy.

The total energy of the system, using the virial theorem and assuming a circular orbit, is:

$$E=K+V=\frac{V}{2}=-\frac{GMm}{2r}.$$

But to have a constant energy over a diminishing mass requires that r diminishes, so I don’t understand how this argument is used to justify an increasing radius.

Answer 1

It is harder to debunk the argument about energy than to explain the phenomenon in terms of forces. As the Sun’s pull (g) decreases, the Earth (in roughly circular orbit at a given radius) find itself going a trifle too fast for its centrifugal acceleration to balance g. The effect on the orbit is the same as if the Sun’s mass had remained constant but the Earth had received a tiny prograde kick. The effect of a perigee kick is to raise the apogee of a satellite’s orbit, but the effect of continuous low-level thrust is to cause the satellite to spiral outward.

Answer 2

But to have a constant energy over a diminishing mass requires that r diminishes, so I don’t understand how this argument is used to justify an increasing radius.

Mechanical energy and angular momentum cannot both be conserved in a variable mass gravitational system. If energy is conserved then $\frac{\Delta r}r = \frac{\Delta M}M$ and angular momentum must change by $\frac{\Delta L}{L} = 2\frac{\Delta M}M$. If angular momentum is conserved then $\frac{\Delta r}r = -\frac{\Delta M}M$ and mechanical energy must change by $\frac{\Delta E}{E} = 2\frac{\Delta M}M$. So which of two, if any, is the case?

There's no compelling reason to invoke conservation of mechanical energy here. While conservation of energy is a law of classical mechanics, conservation of mechanical energy is not. Moreover, even energy is not conserved in this situation as the energy created by nuclear fusion at the center of the Sun eventually radiates into empty space.

On the other hand, that energy transfer into empty space is isotropic and unpolarized; it does not carry angular momentum: Assuming angular momentum is conserved is very much justified.

That mass is being lost and that total mechanical energy is negative means the mechanical energy of the Sun-Earth system increases while its angular momentum remains constant. It would appear that this is "free energy". It isn't. The gain in mechanical energy is many orders of magnitude smaller than is the amount of energy created by fusion (and eventually lost to empty space). In particular, $G \Delta M_s M_e/r \lll \Delta M_s c^2$, by a factor of $3\times10^{-14}$.