How far away you can see something in space?

[Kuroneko]

Trying to figure it out myself, I figure if I take the temperature of one spaceship, plug it into the Stefan-Boltzmann law to get the power radiated, than from there apply an inverse square formula at the distance separating the two ships, I should have the amount of power per square meter (called the power flux, right?) hitting the telescope. Then I can get an energy number from the surface area of the telescope and the duration of the exposure.

This should be fine as a first approximation if this is in interstellar space, although the geometry of the ship would make some radiative directions preferred compared to others. This is, of course, especially true if there is a star nearby.

If that energy number is large enough, it should be visible.

Well, yes, but if the telescope is only responsive in some limited range of the electromagnetic spectrum, as would be more realistic, then one should integrate the Planck law over that range instead.

1) Is my reasoning above correct? If not, what is the right way to do it? Does this also work for the temperature of the rocket fuel being expended?

If the ship is far away from stars, certainly. Inside a star system, one would need to consider reflection along with the thermal radiation.

2) What energy number would be reasonable for a telescope or other sensor, including the naked eye to see something standing out? What emissivity number would be a reasonable estimate for a spaceship or its reaction mass?

Well, that's a bit complicated. Assuming the only radiation source is thermal, take the Planck law, multiply it by the luminosity function exp[-(λ-560.3)²/60.64²], where λ is the wavelength in nm, and integrate over the entire spectrum. The result is the luminous flux in watts; the measure in lumens is 683 times this amount (the same measure one gets on the light bulb packages). This particular model comes from fitting a Gaussian curve on the table in Born & Wolf, and it's actually not particularly good--the peak should be 555nm instead on 560nm (but this makes it easier to use and is more than sufficient for your purposes). Additionally, it is the photopic ("bright light") luminosity curve; I don't have any data on the scotopic ("feeble light") curve, which would be more applicable here, other than its peak being 507nm. The overall shape is nearly identical to the photopic one (from Fig. 4.32 in B&W 3rd ed.). It's probably adequate to simply move the peak from 560 to 507.

One method of comparison would be to calculate the luminosity flux of the sun. Using the above method, except with a cubic rather than quadratic interpolation in logs, the output of the Sun comes out to be 3.3e28 lumens, or 1.2e5 lux (lum/m²) at 1AU, ignoring atmospheric effects. If this corresponds to an apparent magnitude of about -26.7 at 1AU, then an object of apparent magnitude 5 would deliver 2.5e-8 lux; this represents just about the faintest of objects visible with the naked eye, give or take a factor of three, depending on local light pollution, etc. The same analysis would apply to all optical telescopes, except that they are capable of seeing fainter objects.

3) How would this change if there was something in the background, such as if Earth was behind the ship? Would it still be as easily seen?

Of course not. I suppose one way to attempt to quantify it would be to calculate the spectral difference between the background and object, which would automatically account for both raw luminosity differences and color differences. Unfortunately, I'm not familiar enough with photometry to present more than guesses on this matter. Perhaps there is a better way to answer the previous question as well.

Edit: exp[-(λ-560.3)²/60.64²], not exp[-(λ-560.3)/60.64²].

[CaptainChewbacca]

The furthest-imaged superclusters are about 800 million light years away.

[Admiral Valdemar]

If you have a good enough instrument, you can see as far as you want, within reason. Even objects directly in front of others can, if large enough, cause gravitational lensing to highlight the possibility of another object being hidden yonder.

[Sikon]

There are some formulas for the practical range of detection of space warships at Atomic Rocket here, i.e. for infrared. The web page just says the formulas are for "current technology." That is rather vague since additional factors would have to matter like the size of searching telescope arrays, how much of the sky needs to be searched in what timeframe, etc. Still, as they say, the Space Shuttle's main engines could be detected today if it was past the orbit of Pluto, at least with a good telescope looking in the right direction at the right time.

For infrared, the situation might be helped a little with tricks like having the side of a warship facing an enemy be behind a cryogenically cooled miniature "umbrella," while using a mass driver to avoid having hot plasma exhaust. But that would tend to not work for long if the opponent had multiple sensor arrays throughout the star system that could see the ship from more than one narrow angle range. Waste heat has to go somewhere.

There may be two different figures. One figure might be the maximum distance at which a ship could be detected if the most powerful telescope available focused on the right spot of the sky. The other might be the distance at which the ship tends to actually be noticed and detected. As an analogy, there are lots of asteroids being still discovered by amateur astronomers today despite them having telescopes far from the most powerful, and that is partially since the most powerful telescopes aren't spending so much of their time scanning the sky for asteroids.

With that said, a warship would ordinarily stand out like a hot plasma beacon firing its engines, vastly more noticeable than an inert small asteroid. So a 360-degree array of infrared sensors observing the entire sky constantly ought to be able to detect a regular warship rather far away.

A completely powered-down stealth ship might be able to remain hidden for a long time if the sci-fi society had limitations a little like earth today, given that most asteroids below a certain size have still never been detected, though that is much affected by the relatively limited funding devoted to telescopes searching for near-earth asteroids of asteroid-impact risk. The ship firing its engines, weapons, or even just providing life support for an active human crew would make effective stealth harder.

[Gil Hamilton]

Keep in mind that telescope resolution is useful too. One way they boost the resolution of telescopes is to put a bunch of them in an array and link the telescopes together to make what is called an interferometer. An example of this would be the CHARA array outside of Los Angeles, which is six 1 meter wide telescopes mounted in a circle 400 meters wide. Linking these telescopes together and using interferometry, it is effectively like they stuck a telescope 400 meters wide on that mountain, with immense resolution.

Conceivably, if your spaceships wanted to look really really far away with some accuracy, you could have them able to deploy broad arrays (few kilometers across) of high powered telescopes and have them linked together via a central to simulate a Big Fucking Telescope that could resolve details from very very far away. Or given your orbiting satillites, it would be too cool to have a system of tens of thousands of them in orbit moving in intricate patterns that allow hundreds of them at a time to form interferometers on command.

[Ted C]

See if this helps you any...

BabTech Viewing Distance Calculator