Could a planet be habitable due to illumination from the galactic core?

Question

I recently saw this answers where it was suggested that the galactic core could illuminate a planet suffiently to make sure it is always summer on one side of the planet. This got me curious and after some research I came across this.

Astronomers have estimated that the total luminosity of the central dozen or so light-years of our galaxy is equal to about 10 million suns. That sounds big and bright – until you recall that the center of our galaxy is 25,000 light-years away. At that distance, if interstellar dust weren’t in the way, our unaided eyes would see the central part of our Milky Way galaxy as a central glow no bigger than the planet Venus, and no brighter than one of the stars of the Big Dipper. Interesting … but not blazingly bright. But wait. There’s more. The fact is that interstellar dust obscures more than just this central region. It also dims the light of billions of foreground stars, as well as stars surrounding the core itself. If there were no dust between us and the galactic center, the light of all the stars located toward the galaxy’s core would easily exceed that of a full moon. If you looked in that direction, you wouldn’t see much else but the combined glow of billions of stars.

While the first part of the article gives a value of luminosity, the second part points out why one should ignore it.

So given the fact that the galactic center is quite bright, yet itself a terrible place for planets due to gravitational interaction between stars, is there a habitable zone around the galactic centre in which a planet on an orbit inclined enough that it won't be shadowed by the discs dust clouds could be habitable?

I know that when the planet will invariably cross the galactic disc it will receive much less light and have an ice-age, but this is beyond the scope of the question. I would actually prefer if formulas were given in the answers instead of a simple yes or no.

Answer 1

Well, let's do some back-of-the-envelope calculations and see where our intuition gets us. The sun gives off about 3.846 x 10^26 watts of power. At our distance from the sun, we get about 1360 watts per square meter of solar energy intensity. Let's be generous and round the sun's intensity up, and the needed solar energy for a habitable planet down, so we get 4 x 10^26 watts for the sun and 1000 watts per square meter of intensity. Multiply the figure for power by 10 million and we get the total power output of that central dozen-or-so light years of the galactic core, which comes out to about 4 x 10^33 watts. That's a lot of energy. The question is: Can a planet be outside that radius and receive enough energy to survive?

Answering that is pretty simple. Just find the surface area of a sphere 12 light years across, which turns out to be 4.05 x 10^34 square meters. Divide 4 x 10^33 watts by 4.05 x 10^34 square meters, and we end up with... 0.988 watts per square meter; less than a thousandth of what we want to support life. And this is just at the perimeter of that brightest point in the galaxy.

I'm not sure how I'd go about calculating the total power output of the entire galactic core, but my intuition tells me that if the brightest point in the galaxy can't muster the necessary intensity to light up a planet that's practically inside of it, I don't see how the much-less-dense galactic core will be able to do better to a planet that is outside the galactic core altogether.

Although, going back to orbiting the 12-light-year area at the center of the galaxy, I suspect that you COULD make a habitat that could survive off of the radiance of the galactic core. It would need some truly gargantuan solar collectors, but considering that those can be mirrors made effectively out of aluminum foil, it is within the realm of possibility. Perhaps the planet itself could have an gigantic array of mirrors orbiting it, gathering the energy from an area a thousand times larger than the cross-sectional area of the planet itself, and bouncing it down to the surface. Again, though, this would require the planet to be within the bounds of the galactic core, a mere six light years from the center of the galaxy.

EDIT: Actually, I take it back, sort of. The total power output of a region of space might be approximated by the total number of stars in that region of space. The total number of stars could be approximated by the volume of that region of space. The volume of an sphere increases with the cube of the radius of that area, while the surface area increases with square of the radius. With this in mind, if the central 12 light years of the galaxy was expanded out to 12,000 light years, with the same density of stars, then the power output would be a billion times greater while the surface area of the region would only be a million times greater, giving us close to the intensity we were looking for... except that a volume of this size can only be approximated as a point if you're pretty far away. If you're right up next to it, you aren't going to get anywhere near that much power, so it's still a bust.