Can mirrors on the moon allow solar panels to operate at night?
You may have looked up in the night sky before and wondered "Where'd the Moon go?"
The lunar cycle is about a month. Since it's not quite a month, we'll call that a moon-month. There's a less fun, more astronomical term for it that I won't be using. Over a typical moon-month, the Moon will appear half-lit or less for about half of the time, and not always the same half mind you. And at the start of a moon-month, it will be just completely dark. Not great.
There is no point on the Moon that will be illuminated consistently every night. You'd have to cover the near-side of the Moon from end to end to get light almost all of the nights, and then you'd still get nothing on a new moon. Bottom line is that instead of idle solar panels you'll have idle Moon mirrors. Think about that before you whip out the checkbook.
Another reason the Moon might be absent is clouds. I'm not sure if a good enough mirror could allow the light to peer through a hefty cloud layer, but that's definitely something you should find out before you invest trillions in Moon mirror technology.
And also frame challenge.
The problem of idle solar panels need not be one. Energy usage isn't constant over a day, nor day to day. On our modern power grids, we always want to produce as much power as we consume. When we have excess energy, we store it. Solar panels that don't work at night may simply be a feature, not a bug. They work and store excess energy by day, and the stored energy is used by night.
I know the Sun is a pretty cool (but actually quite hot) celestial body, and solar panels are certainly useful, but they're also not a silver bullet. And that's fine, we have other ways to generate power, and you should too. We call that the energy mix.
Now if you're specifically looking for a pretty cool (but actually quite hot) celestial body, available day and night, come rain or shine, to complete your energy mix, you're in luck because there is one. It is called Earth. I'm partial to geothermal energy and I think it is criminally underexploited. I mean, it's just there for the taking, and you barely have to dig for it, and it's good for both electricity and heating.
Anyways, the point is, if your solar panels are down for the night, other power plants can pick up the slack. I'd admit, it's not quite as cool as Mirror Moon, but a healthy energy mix works. And if you're starting to design convoluted space contraptions to lit a few solar panels, you're designing your power grid wrong.