Inside a Spherical Mirror

[Tricky]

The Bean

I don't know if any of you have been to The Bean in downtown Chicago. It is an amazing sculpture and very reflective both inside and out. As you can see from the outside picture, the bean has a little concavity that you can walk inside. The second picture is taken from the inside.

I have more if you are interested.

 

[Stellafane]

He would receive no more energy than he would if he shone the light directly on himself. The sphere only reflects energy, it does not amplify it...

This is of course correct. I think some may be confusing the situation described in the OP with that of an astronomical telescope, which as is well-known can take a pinpoint of light in the sky (often so dim it's invisible to the naked eye) and make it dazzlingly bright. But the reason a telescope can do this is because there's a lot of starlight falling on every square inch of Earth, and the telescope grabs a sizable patch of it and concentrates it into a tiny point. The total amount of theoretically available starlight is quite high because the original source of the light is extremely large and bright, and even at stellar distances produces a considerable amount of illumination falling on the Earth. The flashlight, on the other hand, produces a very limited amount of light. So even if you get every photon of light bounced back to you, it can never be brighter than the original flashlight bulb -- visible, but certainly not a blinding sheet of illumination.

 

[Dorfl]

As I understand it, he wouldn't be seeing anything but a glare from any direction - the flashlight is the only emitter of light, whereas his body etc. only reflect - this would be swamped by the continuous emission from the flashlight.

(Just tried to model that in a Bryce scene, with a light source and a stone texture cube inside a perfectly reflective sphere. Sure enough, once you enable total internal reflection to a high degree (obviously infinite is not available, but a couple hundred iterations can be done in acceptable time), it's all shapeless glare. I wouldn't treat that as any great evidence obviously, although Bryce's laws of optics are pretty good.)

Hm, I didn't think of that. Intuitively I assumed that looking in any direction, he would see either a light beam coming from his own body, or one from the flashlight. So at worst his view would be divided into bright fields from the flashlight, and dim ones from himself. But if you say Bruce gives a uniform glare, I suppose my picture must be wrong.

What if the astronaut himself were the light emitter? (Maybe Doctor Manhattan created himself a mirrory sphere, just to see what would happen.)

 

[Cuddles]

If we assume the flashlight uses two size D batteries and they each contain 75,000 joules of energy ( http://www.allaboutbatteries.com/Energy-tables.html we'll assume alkaline batteries although I'm sure NASA uses some crazy alien-technology, area 51 type battery. Probably Lithium-ion.) that would be some 150,000 joules together.

http://www.engineeringtoolbox.com/human-body-specific-heat-d_393.html says the average human body has a specific heat of 3470 J/kgC. So if we ignore his gear and flashlight, and assume he is 70 kilograms ... his temperature will go up by about .6 degrees C. Not enough to kill him. We need bigger batteries.

I don't think that's correct. You're assuming that his body simply absorbs all the energy. However, at the same time it will be acting as a black body emitter. In fact, the astronaut is likely to be by far the biggest radiation source present. Assuming he has enough food, he'll eventually die once the inside of the sphere has heated up past 37^o and he can no longer regulate his body temperature.

 

[dogjones]

I bought some of those photons once. They're not nearly as good as fresh photons. You should always buy your photons from a reputable source.

Maudsley does a good line.

 

[Unlike a Bull]

I don't think that's correct. You're assuming that his body simply absorbs all the energy. However, at the same time it will be acting as a black body emitter. In fact, the astronaut is likely to be by far the biggest radiation source present. Assuming he has enough food, he'll eventually die once the inside of the sphere has heated up past 37^o and he can no longer regulate his body temperature.

You are correct that I am incorrect. I was just thinking about worst case (hypothetical) scenario where all the energy from the flashlight is converted to heat and all is transferred to his body. If this worst case scenario won't kill him then the lesser, realistic ones won't either.

I'm not sure if his metabolism would be enough to kill him either though. It is specified that the sphere reflects light perfectly, but nothing is said about it's heat conductivity. Therefore we're free to speculate! If the sphere is a perfect insulator, then yes his metabolism will eventually kill him. If, however, the sphere allows heat to be released to the surroundings then it's just a matter of how hot the thing will get before equilibrium is reached.

also, I don't think he would lose his ability to regulate temperature when the sphere reaches 37 C. I know he would lose his ability to radiate heat to his environment, but he could get creative. Once he started getting warm he would start sweating. He would be able to regulate temp through sweat for a while, but the air in his suit would become warm and humid pretty quickly. But, by venting this humid air into the sphere, he would be removing heat from himself and putting it into his environment. This air would then be replaced by whatever kind of breathing system he had. He would be able to do this for a while, since all he would need is a pressure difference to vent the air.

please note, I'm not saying that a 37 C astronaut could make his surroundings greater than 37 C, but by moving mass from himself to his surroundings, he can remove heat from himself.

 

[Cuddles]

If, however, the sphere allows heat to be released to the surroundings then it's just a matter of how hot the thing will get before equilibrium is reached.

please note, I'm not saying that a 37 C astronaut could make his surroundings greater than 37 C, but by moving mass from himself to his surroundings, he can remove heat from himself.

Well, I assumed that since we were dealing with an astronaut that he'd be floating in a perfect vacuum. If you allow atmosphere inside then that changes things a bit, and by playing with the conditions inside and outside the sphere you can come up with pretty much any outcome you like.

 

[Unlike a Bull]

Well, I assumed that since we were dealing with an astronaut that he'd be floating in a perfect vacuum.

I'm operating under the same assumption (well, maybe not a PERFECT vacuum). I meant that the sphere would be able to radiate heat away from itself. When I talked about its conductivity I meant it's ability to transfer heat from it's inside edge to it's outside edge so that the energy could be radiated away, not that it would be conducting heat into some medium outside of it. Sorry if I wasn't clear enough.

If you allow atmosphere inside then that changes things a bit

Why wouldn't you allow atmosphere in the sphere if it's vented from the astronauts suit?

and by playing with the conditions inside and outside the sphere you can come up with pretty much any outcome you like.

Pretty much.

This is because we weren't given enough information in the OP, because the OP was about what the astronaut would see, not how hot he would get.

 

[Cuddles]

I'm operating under the same assumption (well, maybe not a PERFECT vacuum). I meant that the sphere would be able to radiate heat away from itself. When I talked about its conductivity I meant it's ability to transfer heat from it's inside edge to it's outside edge so that the energy could be radiated away, not that it would be conducting heat into some medium outside of it. Sorry if I wasn't clear enough.

But if there's a vacuum inside the sphere then it can never absorb any heat. It's a perfect reflector, remember, so it can't absorb any radiation. An atmosphere would allow conduction, but if you get rid of that there's nothing.

Why wouldn't you allow atmosphere in the sphere if it's vented from the astronauts suit?

Because, assuming the question was about what would happen in a vacuum, it would be changing the question. It's like starting with the question asking about a perfect reflector and then saying that the astronaut could throw something at it to break it. That sort of thinking is great for silly lateral thinking questions, but is utterly pointless when applied to thought experiments trying to answer a specific question.

For my part, I wouldn't allow atmosphere in the sphere if it's vented from the suit because I was thinking about what would happen in a vacuum. As I said, if you allow an atmosphere then you can come up with pretty much any result depending on what assumptions you make, but in that case it makes no difference where the atmosphere comes from, so it's completely pointless to wonder about different cooling systems the astronaut might have.

 

[Unlike a Bull]

But if there's a vacuum inside the sphere then it can never absorb any heat. It's a perfect reflector, remember, so it can't absorb any radiation. An atmosphere would allow conduction, but if you get rid of that there's nothing.

Oh! For some reason I was thinking perfectly reflective only in regards to visible light. I see my error now. Thanks.

 

[Dan O.]

Don't look at the first reflection directly though, although if my scenario is correct, the only way that could focus directly on the eyes is if he was holding the flashlight between his eyes.

The first reflection will create a somewhat distorted and blurred image at a point that is an equal distance on the opposite side of the center of the sphere. The closer that the source is to the center, the sharper the image will be. If the astronaut is at the location of the first image, light would be seen coming from all directions except where the astronaut's own shadow blocked the light.

The intensity of the light at the image will follow the inverse square law with the distance measured from the image not the original source. The maximum intensity at the image cannot exceed the intensity of the source. However, since the image is not protected by a bulb, it is possible to get much closer to it.

If the astronaut is closer to the center than the source and on the line through the source and the sphere's center then the astronaut will not see the first image of the source except for some extra light coming from directly behind the source. The astronaut's own shadow will be blocking the light that would have created the image.

Anything inside the sphere that is illuminated (such as the astronaut) will be a secondary light source and will also produce an image opposite the center. The astronaut will see an image of himself but without the reversal that you see in a mirror. Also unlike an image produced by a flat mirror, it is possible to pass through this image by passing through the spheres center.

 

[Dorfl]

The first reflection will create a somewhat distorted and blurred image at a point that is an equal distance on the opposite side of the center of the sphere. The closer that the source is to the center, the sharper the image will be. If the astronaut is at the location of the first image, light would be seen coming from all directions except where the astronaut's own shadow blocked the light.

The intensity of the light at the image will follow the inverse square law with the distance measured from the image not the original source. The maximum intensity at the image cannot exceed the intensity of the source. However, since the image is not protected by a bulb, it is possible to get much closer to it.

If the astronaut is closer to the center than the source and on the line through the source and the sphere's center then the astronaut will not see the first image of the source except for some extra light coming from directly behind the source. The astronaut's own shadow will be blocking the light that would have created the image.

Anything inside the sphere that is illuminated (such as the astronaut) will be a secondary light source and will also produce an image opposite the center. The astronaut will see an image of himself but without the reversal that you see in a mirror.

I have difficulty visualizing that, but it does sound cool. Thank you very much, I had been wondering about it for a few years now.

Also unlike an image produced by a flat mirror, it is possible to pass through this image by passing through the spheres center.

[Prince of Persia] Sheathe your sword and pass through your mirror image... It took me weeks to figure that out! [/Prince of Persia]

 

[Timothy]

Isn't this just an integrating sphere?

No, an integrating sphere relies on diffuse reflection at each bounce to eventually produce uniform illumination. This is specular reflection.