Can astrophage actually absorb energy in space ?

I'm trying to think about the energy absorption and expenditure of astrophage. I understand it can absorb heat, so it would work on earth, but in space, it doesn't take heat from the sun, right ? Just radiation ?

So, let's imagine an astrophage in a circular orbit around the sun, getting no closer or further away (yes, I know they don't do that). So, it's absorbing radiation from the sun. But doesn't everything it absorb mean it gets a push away from the sun ? And so to stay at the same distance, it would need to push that same amount of energy out away from the sun ? So, then how is it getting any net energy ?

Now, in "reality", it's actually traveling towards the sun in an arc, and away again, and not apparently using orbital mechanics (that a true solar sail can use to lower its orbit). So it would use energy to get to the sun, and maybe absorb some while cruising back to the planet.

Am I missing something about the energy equation here ? Are the astrophage somehow able to absorb a photon's energy but not its momentum, yet when it creates a photo, it is affected by the momentum ?