Other Foreign Skies

This post is a response to a question I got on my Ask Tsana page.

Sam Keola asked:

Love the views of Jupiter from Ganymede and Io. How large would it appear from Europa or Callisto? And how large exactly would the sun appear? (I know tiny as hell, but another lovely picture would be amazing.)

The mathematical answer to that is explained in this old post. And my first set of Jupiter images (Io and Ganymede's skies) can be found here.

Jupiter

This time around, I used a different image of Jupiter so if you're wondering why it's rotated relative to the old pictures, that's why. For the Jovian images, I've used the same starting image because in the year since I last did this, I haven't managed to take a more suitable photo. Such is life.

The original photo with a full moon in Earth's sky.

So. Europa is the second Galilean moon out from Jupiter. It's made mostly of ice, is the smallest of the Galilean moons and might harbour life in its subsurface liquid ocean. The diameter of Jupiter as it would appear in the Europan sky is almost 24 full moons across. Remember that Europa's sky wouldn't actually look blue either since it doesn't have an atmosphere but I don't have a decent night skyline to work with. I'll do a night version eventually.

The size Jupiter would appear in Europa's sky. Or in Earth's sky if you swapped it with Europa.

You might be wondering whether Jupiter would actually be oriented the way it appears in these images. Well it depends. The direction the bands run relative to the moon's horizon would depend on where on the moon you were. Close to the equator, the bands would be vertical (although if Jupiter was high in the sky, it would be pretty difficult to tell. Perhaps better to say east-west). If you were near a pole, they'd be horizontal as in these images. And remember, the Galilean moons are all tidally locked, so Jupiter would never move, just change how much of it was illuminated by the sun.

And Callisto, the most distant of the Galilean moons. Callisto's Jupiter would appear "only" about 8.5 full moons across.

The size Jupiter would appear from Callisto. If Callisto had an Earth-like atmosphere and gum trees.

The Sun
 
The second part of Sam's question was how large would the sun appear from Jupiter. Well, on Earth, the sun and the moon appear to be approximately the same size (there's a little bit of a difference when the sun is at its closest and the moon at its furthest and vice versa). So the sun from Earth is about one full moon in diameter.

From Jupiter (or its moons) the sun would appear about 0.4 full moons across which is a little bit less than a sixth of the area of the sun as seen from Earth (remember, the moon and sun seen from Earth are on average the same size).

I cheated a little bit with these next two sun photos. They're actually two separate photos and I made the sun smaller in one of them. The reason the rest of the photo looks darker for the Jovian sun is because I was fiddling with settings on my camera. And if you're wondering why I chose sunsets, it's because those (and sunrises) are pretty much the only kinds of photos where the disc of the sun is properly visible.

Ordinary sunset on Earth:

Sunset. A little bit more than half the sun is below the horizon.

Sunset if Earth was at the same distance as Jupiter (but yet still warm enough to have liquid water. And plants. By the way, with these two, it's probably clearer if you click on the images to enlarge and compare the sun side by side.

A more diminutive sun, less than a sixth of the area of Earth's.

And there you have it. Photoshopped images (well, actually, I used Pixelmator) depicting the sizes of Jupiter and the sun from the Galilean moons and the Jovian system, respectively.

Stars in their Skies

My intention had been to write another gravity post, this time about tides, tidal locking and Saturn's rings, but unfortunately I've been too busy to fully cover the scope I wanted to. Instead of a half-hearted post on the above, I bring you: stars! Stay tuned for tidal forces next week.

It's full of stars!

There are many different types of stars, as you may recall from the HR diagrams I've discussed previously (I've also drawn a rather crude, annotated HR diagram below). Stars come in a whole spectrum of colours, based on their (surface) temperatures, and these are loosely correlated with their masses. In general (for the main sequence), small stars are red and hence cool (only 3700 Kelvin) and large stars are blue and hence very hot (more than 33 000 Kelvin).

These large blue stars are sometimes called blue giants and small red stars are often called red dwarfs. In terms of mass, these stars can range from over 100 times the mass of the sun for blue giants down to about a tenth of the mass of the sun for red dwarfs. The sun, by the way, is on the cooler and smaller end of the middle of the main sequence.

Aside from descriptive words, stars are also separated into lettered classes based on their temperature. The hottest stars are designated O type, then it goes B, A, F, G (the sun is a G-type star), K and the coolest red dwarfs are M type. I'm not terribly fond of the mnemonic I use to remember them, so if you can think of a good one, please leave it in the comments. ;-)

Main sequence stars generate energy by fusing hydrogen and helium in their cores. They need to generate enough energy to over come the pressure of gravity trying to pull them into their central point. This means that bigger stars burn their hydrogen faster because they need to produce more energy to prevent gravitational collapse. Well, technically it's increased gravitational pressure that drives faster fusion in the core. Smaller stars don't have as much gravitational pressure acting on them, so the hydrogen nuclei in their cores are pushed as close together, meaning that their fusion reactions happen much more slowly and it takes longer for them to burn through all their fuel. So big stars burn bright and fast and die young. Small stars burn more conservatively and lead much longer lives. For reference, the lifetime for the hottest blue stars is around a million years whereas red dwarfs can expect to live a lengthy fifty billion years or so (the universe is currently only 13.4 billion years old). Our sun has been around for four and a half billion years and can expect to keep going for another five billion or so.

A very crude representation of where different types of stars fall on the HR diagram.

Stellar life

Talking about how long stars last is all well and good, but where do they come from and where do they go?

Nebulae (singular: nebula) are giant clouds of dust and gas in space. Even though the particles in a nebula are fairly spread out, over enough time gravity pulls them together into clumps. When these clumps get big enough that the gravitational pressure holding them together is great enough for fusion to start in the core, they "turn on" and switch from warm balls of gas into blazing young stars. The rest of the gas and dust that didn't make it into the protostar before it turned into a star proper either get blasted away by the new stellar wind unless they already clumped together enough to form planets.

After that, the star lands on the main sequence, based on its mass. What happens when it uses up all its fuel and reaches the end of its main sequence life varies depending on its mass. Smaller stars, like our sun, will throw off their outer layers and swell up into a red giant. The star will eventually eject all its outer layers and all that will be left is the star's exposed core; a white dwarf. A white dwarf no longer fuses hydrogen or helium or anything else. Instead it just slowly radiates away all of its heat until it eventually (over trillions of years) cools. White dwarfs are basically just spheres of carbon or oxygen or a mix of the two, depending on the initial star. Those news stories you might have seen about the "largest diamond in the universe"? Those are talking about carbon white dwarfs.

If we start with a larger star—big enough that after the star has ejected all its outer layers the core that's left behind is more than 1.4 times the mass of the sun—then the core will be too massive to remain merely a white dwarf. Instead it will collapse in a giant explosion known as a supernova. In the immense pressures exerted in the supernova, the protons in the old stellar core combine with the electrons to form neutrons (atoms are usually made of protons, neutrons and electrons). This type of star, composed entirely of neutrons and not found on an HR diagram, is called a neutron star. A really massive star can, after a supernova, collapse into a black hole, an even denser object (previously mentioned here).

Life elsewhere

Since the sun is a G type star and has a life-bearing planet, of course G type stars elsewhere could have life-bearing planets too. There is also a reasonable chance that F and K type stars, which are fairly similar to our sun, could also harbour life. The biggest problem is when we look at very massive stars. When we're talking about stellar lifetimes in the millions of years, then there probably isn't enough time for life to form. It took four or so billion years for life on Earth to get to the stage it is now. If the sun had only lasted for ten million years before exploding, then we wouldn't be here.

What about red dwarf stars then? They live for a very, very long time, so there's certainly ample time for life to develop. However, because the habitable zone is so close in to the star (because the star is so dim and cool), there are two issues:

1. The planet will probably be tidally locked, meaning that one side is always facing its sun while the other never gets any direct heat or light. It's likely in this case that both the day side and the night side will be permanently too hot or cold. The ring of the terminator (the boundary between the day side and the night side) would probably be the best bet for life developing, temperature-wise.
2. Small stars seem to have more flares than larger stars. Flares aren't terribly conducive to life, particularly at such close proximity. This is still an active area of research, however.

But, theoretically, in optimal conditions life could develop on a planet orbiting a red dwarf. Also, both of those issues are things I will probably blog about in the future.

As I mentioned last week, just because life can't develop there, doesn't mean humans can't try to colonise planets around different and interesting stars. Of course, colonies with a five or so million year time-limit might be a bit sort sighted, but if there are planets in suitable places, it could be good for a laugh.

The humanly habitable zone

If you want your little book people (otherwise known as characters) living on the surface of a planet that isn't Earth and without constant artificial support, you probably want the planet to be habitable. This is where the habitable zone (and hence this blog post) comes in.

Why is it a zone?

First let's talk about what we need from a planet. Earth is great; it gives us lots of handy life-sustaining conditions; air, water, the right amount of gravity, sunlight and radiation... Of course, the reason the Earth is so well-tuned to keeping us alive is that we evolved on Earth. It's not tuned to us, we're tuned to it. There is no reason that aliens couldn't evolve in very different conditions to those found on Earth. Even on Earth there are many forms of life such as extremophiles which live in conditions we humans couldn't survive in. There's also a good chance that life is possible on Europa or Titan (moons of Jupiter and Saturn respectively), but on the former it would have to be in a subsurface ocean and the latter has a very different atmosphere and composition to Earth, meaning that life couldn't be water-based.

Aliens are all well and good, but if we're interested in what conditions humans can live unsupported on a planet's surface, we need to be much more specific.

The things I mentioned earlier (air, water, heat, etc) depend on two things:

1. Planetary properties such as size and composition, and
2. Planetary location.

The first of those is most easily summarised as having to be similar to Earth to sustain unaided human life. The latter is the subject of this blog post.

You see, although the planet needs to be similar to Earth, its sun doesn't have to be that similar to our sun. Different stars output different amounts of light and energy (previously discussed in terms of how bright they are). If we were to pick Earth up and put it in orbit around a larger, hotter star then, if we didn't want to be burnt to a crisp while all our water boiled, we would have to put Earth in a further out orbit to compensate for the extra energy coming off the star. Similarly, if we put Earth around a smaller, cooler star, we'd need to put it in a closer orbit to stop it turning into a snowball.

The optimal region in which to put a habitable planet is called the habitable zone. It's a zone because there's no reason for life to have not evolved on Earth if it had been slightly closer or further away from the sun. I'll get back to life evolving on other planets at the end of the post, though.

There are a few different ways of defining what constitutes a habitable zone. Some definitions involve the region in which liquid water is possible, temperature-wise (since our biochemistry depends heavily on liquid water), other definitions go a bit further and include things like carbon cycles and the greenhouse effect (pdf link, sorry). For the purposes of the habitable zone calculations which follow, I'm going to use the definitions given in Selsis et al. (2007).

Before that, though, let's work out how warm a planet is based on how far from its sun it is. I am using a formula adapted from a Melbourne Uni 3rd year physics lab manual because it's simpler than the one given in Selsis et al. (2007), although they both give the same results. The approximate temperature of a planet, given a stellar temperature and radius and ignoring complex atmospheric effects (but assuming that an atmosphere exists) is:

T is the approximate temperature of the planet in Kelvin, T_* is the temperature of the star in Kelvin, A is the albedo of the planet (for reference, Earth's is 0.306, according to wiki), R_* is the radius of the star and d is the distance of the planet from the star.

The only tricky thing here is that both R_* and d have to be in the same units, so either both kilometres, or both AU etc.

According to Selsis et al. (2007), life is possible in the range where the temperature is between 277 K and 394 K. Rearranging the above equation then with these values and substituting R_* in annoying units for solar radii (the HR diagram I linked to a few weeks ago can help you estimate this much more easily than in km or AU), we find that the habitable zone for any main sequence star in AU is given by:

Quantities as above. R_☉ is solar radii (multiples of the radius of the sun) and d now must be in AU.

So all you have to do now is work out what kind of star you want, find out it's general properties (easily googleable if you're using a real star) and throw in the numbers. Unless you have plans to make your planet unusual, it's probably best to keep A as the Earth's albedo. That said, habitable distance will change with albedo so the only reason not to change it is because albedo reflects the planetary composition (and we probably want to keep a similar composition to Earth...).

For reference, Selsis et al. (2007) find the sun's habitable zone to be between 0.95 and 2.4. (Note that they use a different albedo for the Earth.) Wiki lists some other numbers which seem to vary mostly in the outer edge prediction.

I should also point out that all of this assumes that there aren't any other stars near our planet. Things get a bit more complicated with multiple stars around, something I will definitely address in a later post.

Evolving elsewhere

As well as the habitable zone, there is something known as the continuously habitable zone (or CHZ). This is the region around a star which remains habitable throughout the star's (main sequence) lifetime. (Main sequence lifetime because different stars end their main sequence lives in different ways, mist of which make the continued survival of planets complicated at best. More on this in a future post, I think.) The reason we need to worry about different times in a star's life is that stars tend to increase their rate of energy output as they age and use up fuel (because as more fuel is used up, hotter core temperatures are required to sustain fusion which leads to an increase in luminosity). The CHZ is narrower than the habitable zone calculated at any given time during the star's lifetime and it is possible that as the star's luminosity changes, the habitable zone could shift so that the planet moves into or out of it over (astronomical) time.

Of course, this doesn't matter so much if we just want to find a planet to throw humans at. Continuous habitability becomes more relevant when we're talking about life evolving. Evolution from scratch takes a long time, but human civilisation to date has lasted an insignificant amount of time, on an astronomical scale.

Gravity: Relatively general space

Two weeks ago, I talked about basic Newtonian gravity. Today I'll be talking about some of the contributions Einstein made.

Curvy

You may have heard in passing phrases like "space is curved" or "the curvature of spacetime", but what do these phrases actually mean? Einstein's theory of general relativity brought us the understanding that the force of gravity is the result of masses deforming (curving) the fabric of spacetime surrounding them. This is different to the other forces of nature which are quantised and mediated by force-carrier particles, and can be mathematically combined into a single force known as electroweak. Don't worry if that last sentence didn't make sense. The important thing is that our deepest understanding of gravity does not include quantum theories* but does involve classical geometry.

The most common metaphor used to describe the curvature of spacetime gives something like this:

Imagine the universe is an infinite rubber sheet (infinite because this is not the place to think about what's happening at the edges or even whether edges exist). Masses such as stars, planets and so forth are like ball-bearings stuck to the surface which, because of their mass, create dips in the sheet. Heavier things make deeper dips. Then, if you roll another ball bearing along the sheet (it doesn't actually have to be a lighter one, but it's easier to picture if it is). As it rolls past the other masses, it's path will be deflected by the dips so that it doesn't go in a straight line from the point of view of an external observer. If you were to draw a grid on the sheet before letting the masses stretch it, then rolled a very small mass very quickly along along it, it would follow the grid lines, even though the grid lines themselves are now stretched.

The very small, light mass I mentioned at the end would have to be a photon, a massless particle of light. Anything with a mass, even a small one, would get deflected off the grid lines by the masses because it would be travelling more slowly that the speed of light.

Of course, the universe isn't a two-dimensional rubber sheet and this metaphor isn't perfect. It's a little bit harder to picture in three dimensions, but the qualitative ideas are the same. And you can think of the grid lines as geodesics which mean they represent the shortest distance between two points and hence the path light takes. That's right, gravity curves the path light takes. The larger the mass, the greater the curvature. This is called gravitational lensing and is all sorts of useful in different areas of astrophysics.

Now, Newtonian gravity doesn't take the curvature of spacetime into account. Usually this doesn't matter much because even the sun only causes enough curvature for Mercury, the innermost planet, to really be effected. The general relativistic (GR) corrections for the other planets are small enough to be insignificant. GR becomes much more relevant around small and dense objects such as neutron stars and black holes.

* miscellaneous proposed but untested theories notwithstanding.

Black as black

Black holes are an interesting concept that falls out of general relativity. Chances are you've heard of them if you haven't lived in an internetless cave for the past hundred years. But what exactly are they?

A black hole is an extremely small and extremely massive object. It is so small and massive that it is denser than any form of matter we know of. We don't really know what sort of matter black holes are made of. This comes from the fact that we don't have a theory of quantum gravity as I mentioned earlier.

Escape velocity is the speed you need to go to escape a body's gravitational pull. To completely escape Earth's gravitational field, for example, you need to leave the Earth at about 11 km/s which is about forty thousand kilometres per hour. To escape the sun's gravity from the surface of the sun (let's ignore the fact that you'd be fried while you were there) you need to leave at about 618 km/s or more than two million kilometres per hour. To escape the sun's gravity from a distance of 1 AU (the distance between Earth and sun) you need a velocity of 42 km/s (150 thousand km/h) because the force of gravity drops off with the square of distance.

Because black holes are so dense, there is a point where the force of gravity is so strong that the escape velocity is equal to the speed of light. This is called the event horizon and from within the escape horizon nothing can escape the gravitational pull of the black hole (since nothing can move faster than the speed of light). This is where the "black" part of "black hole" comes from.

In practice, it would be very hard to escape the black hole long before you reached the event horizon, purely due to the energy needed to overcome gravity. And even before you reached the event horizon (also known as the Schwarzschild radius), lots of strange and interesting things start happening. Any signals you try to send out (to people further away from the black hole) would be redshifted as the wavelengths of light get "stretched out" by the extreme curvature of spacetime around the black hole. Notice how "spacetime" includes the word "time" as well as "space"? Time also gets stretched out near a black hole and passes more slowly (for complicated reasons I might explain in a future blog post). Not that you would necessarily notice if you were falling into a black hole because the tidal forces would be ripping you apart.

Tidal forces come from the difference in the force of gravity between the end of you/your spaceship closest to the black hole and the end further away. Gravity acts more strongly on the closer end, causing interesting (and painful) things to happen. This is the same principle which leads to moons being tidally locked in their orbits around their planets, but taken to an extreme scale. If the force of gravity on your feet is appreciably stronger than on your head, nothing pleasant will result from your feet being sucked into the black hole more quickly than your head.

In terms of where we can find black holes, that's a good question. The only black hole whose existence we're sure of is the supermassive black hole at the centre of our galaxy. We're also quite confident that most other galaxies also contain central black holes. The supermassive part relates to the fact that they are hundreds of thousands times more massive than the sun. In face, our supermassive black hole, Sagittarius A, is about four hundred thousand times the mass of the sun.

Theoretically, there should also be much smaller black holes around, only a few times the mass of the sun, ranging up to a hundred or so solar masses. These would come from very large stars which reached the end of their lifetimes, went up in a supernova and then collapsed into a (stellar) black hole. Even though there's no reason for these to not exist, we have yet to decisively detect any. The black part of the black hole makes that a little tricky. (The is some radiation coming off them, known as Hawking radiation, as well as some interesting things happing as things fall into black holes, which gives us some candidates, but as far as I know they're still only candidates.)

Barely scratching the surface

So that, very briefly, is what general relativity and black holes are about. You might have noticed that there weren't any equations in this post. (Gasp!) That is because most of maths that describes general relativity requires a maths or physics degree to understand. In my experience, general relativity is usually a graduate level subject (or Honours at most Australian universities).

Maths aside, I feel like I've barely scratched the surface of black holes [insert bad pun here], so I suspect another post focussing on black holes will happen some time in the future.

Living on a moon: How bright is the night?

Let's say you've stuck a colony on the moon of a gas giant. I've already talked about the unusual way in which the sun and the primary planet move (or don't move) across the sky. As you might recall, there will be times in the moon's orbit when, depending on where you are on its surface, the only natural illumination comes from its primary planet. the question this post addresses is: just how much illumination can we expect?
There are two things we need to know to work out how much illumination the primary is giving the moon:

1. How bright and far away is the sun?
2. How reflective is the primary?

EDIT: I've added in some comparisons with light bulbs thanks to Patty Jansen pointing out that the human eye can adapt to see in lighting conditions much dimmer than the sun

Star light, star bright?

The amount of light that reaches your planet-moon system from its sun will depend on what kind of star it is. Stars come in different sizes and different temperatures. Most stars lie on what is known as the Main Sequence. Two notable exceptions are red giants and white dwarfs. The main sequence refers to the band of stars running diagonally through the Hertzsprung-Russell Diagram (HR diagram, previous links to two different images). Is basically a plot of how much light a star gives out (it's magnitude or luminosity) against it's colour or temperature. Stars are then divided into types (O, B, A, F, G, K, M) based on colour/temperature. Giant stars (other than blue giants) lie above the main sequence and white dwarfs lie below it. The sun is a G type star with temperature 5800 K (on the surface, that is; it's much hoter on the inside). K means Kelvin and is the standard unit of temperature. To convert from Kelvin and Celcius, you need to subtract 273, so the sun is 5500ºC (with rounding).

Using a star's temperature we can work out how much energy, in the form of light, reaches our planet. The first step is to assume that the star is a black body. This might sound conter-intuitive since the last word you're likely to use to describe the sun is "black", but from a physics perspective, a black body is something that absorbs all incident light and emits light based on its temperature. Well, I say "light", but really I mean electromagnetic radiation.

The Stefan-Boltzmann law tells us how much energy a black body emits based on its temperature. When we're talking about stars, this is called the luminosity. The formula for calculating luminosity is:

A stars luminosity, given it's radius, R, and temperature, T. σ = 5.67 × 10^-8 is the Stefan-Boltzman constant and π = 3.14

The temperature has to be in Kelvin and the radius in meters to give luminosity in units of Watts (yes, like your light-bulbs) which is a measure of energy emitted per second. Radius and temperature are slightly trickier to come up with numbers for. If you're using a real star, you can just look it up on Wiki or Wolfram Alpha (Wiki even has a page listing the nearest stars to Earth). Otherwise you can make up a star with the characteristics you want such as temperature or class, then go to the second HR diagram I linked and look at the diagonal lines of radius. Whether you want a main sequence star, white dwarf or giant, this should give you an idea of radius (in units of the radius of the sun).

That's all well and good, but what we actually want to find is the light reaching a planet, not the total light emitted. Because stars emit light in all directions at once, their total energy output end up being diluted over an expanding sphere of light. Basically, not all the energy the sun produces hits our planet. It depends on how far away the planet is. This next formula will tell us how much energy hits the planet:

P is the energy per second hitting each square meter of the planet and D is the distance from the planet to its sun.

So P is the energy from the sun that hits a square meter of a planet which is D meters away from the sun. We're not quite there yet, but let's take a break and calculate some numbers. I'm going to work out the energy from the sun that hits the Earth/moon and Jupiter each second.

• Earth/moon are about 1.5 × 10¹¹ m from the sun. The sun's radius is 6.955 × 10⁸ m and its temperature is 5800 K. The energy hitting a square meter of the Earth or moon each second is 1400 Joules.
• Jupiter is 7.8 × 10¹¹ m from the sun. The energy hitting a square meter of Jupiter each second is 50 Joules, which is about 3.6% of the energy hitting the Earth. Jupiter's greater distance from the sun means that the sun's energy is about 30 times more spread out by the time it gets there. (As I calculate below, this is still about 14000 times brighter than the full moon as viewed from Earth.)

Planetshine

Light doesn't get completely absorbed by the planet, however. Some of it reflects back out into space and can illuminate other nearby objects. The property which determines how reflective something is (in this context) is called albedo. The average albedo of a planet is a number between 0 (non-reflective) and 1 (absolutely reflective), which represents the percentage of incident light that will be reflected.

In practice, it's fairly easy to implement albedo. The reflected energy is the incident energy multiplied by the albedo. Just multiply P above by albedo, A, and you get the power reflected off each square meter of planet. You can look up albedos for different planets/moons on Wiki and elsewhere. (But we all know Wiki's the easiest. It lists albedos in the summary box on the right of the relevant page. If more than one is given it's the Bond albedo, not the geometric albedo, that you want.)

What we actually care about, however, is how much of that reflected light goes on to reach the moon our colony is built on. In a way, we just reuse the equations I've already included above. Instead of putting L into the equation for P, use the P from the sun multiplied by albedo, radius becomes the radius of the planet, and distance is now the distance between planet and moon:

P is the energy per square meter per second hitting a moon, A is the albedo of the planet, R is the radius of their sun, r is the radius of the planet doing the reflecting, T is the temperature of their sun, d is the distance between planet and moon, D is the distance between planet/moon and sun. The last line is included because if you're using a real star, luminosity will probably be listed somewhere. Otherwise, the penultimate line is what you need to use.

OK, so this is getting increasingly more complicated looking, but remember that you only really have to do the last step. There rest are only there by way of explanation.

Now, one last thing before I calculate some more numbers. That last equation assumes that the primary planet appears full in the sky. If it's half full, you have to halve that number, if it's a quarter full you have to divide by four. Honestly? Just approximate.

• The moon has an albedo of 0.136. The energy the full moon is reflecting at the earth is 0.0037 W/m².
• For the purposes of comparison, a 100 Watt light bulb from 10 meters away has a brightness of 0.02 W/m².
• The Earth has an albedo of 0.306. The energy Earth reflects at the moon is 0.12 W/m². So because it's bigger and more reflective, the Earth as seen from the moon gives off about 32 times more energy per second. That means the full Earth in the lunar sky is roughly 32 times brighter than the full moon in Earth's sky and six times brighter than a 100 W light bulb.
• Jupiter has an albedo of 0.343. Ganymede is 1.1 × 10⁹ m away. The brightness of full Jupiter in Ganymedean sky is 0.07 W/m². That means Jupiter is almost twenty times brighter than the full moon. Not surprising given how big it is in the Ganymedean sky. A half-full Jupiter would be 10 times brighter than the full moon, a quarter-Jupiter about 5 times as bright and so-forth. The varying quantity here is what fraction of Jupiter's disk is illuminated (and that we're working under the assumption that Jupiter reflects evenly in all directions). A quarter-full Jupiter would be about as bright as a light bulb and a full Jupiter would be as bright as three and a half light bulbs 10 meters away.
• Io is 4.2 × 10⁸ m from Jupiter. The brightness of full Jupiter in Io's sky is 0.48. So Jupiter is shining a whopping 130 times brighter than the full moon. By comparison, the sun as seen from Io is only about 100 times brighter than Jupiter. Light-bulb-wise, Jupiter would be as bright as 24 100W light bulbs 10 meters away.
• For a bit of fun, the brightness of full Io (albedo 0.63) as seen from Ganymede varies from 1.2 × 10^-4 W/m² when it is at its closest point to Ganymede to 2.4 × 10^-5 W/m² when it is at its furthest. Neither of those are very bright, but it would still definitely be visible. It's about 0.6–3% the brightness of the full moon.
• And finally, let's say we put Jupiter at the same distance from the sun as Earth is. Now Ganymede would get around the same amount of energy from the sun per square meter as Earth does and Jupiter would be a lot, lot brighter. How bright? 1.9 W/m², which is 500 times more light that Earth gets from the moon and as bright as almost 100 light bulbs from a distance of 10 meters.

And there you have it. A method for approximating how much light you'd get reflected from a gas giant planet (or whatever planet/moon/asteroid you like). Unfortunately this post ended up being a little bit more complicated than I had initially anticipated (where complicated really means more maths), but it's a small price to pay for painstaking accuracy... Well, some semblance of accuracy, at any rate. There are a lot of approximations in the above (for example, the albedo varies for different types of terrain; so Earth's albedo is higher over clouds than over forest), but on average, it's close enough. Phew!

One last thing I came across after writing this post. I was looking for something else and came across this photo of Jupiter and Io. Notice how the line between Io's sun side and dark side (called the terminator) is very distinct and solid, whereas Jupiter has a bit more of a gradient going from light to dark? This is because Jupiter has an atmosphere (a very thick one, but the effect applies to Earth's atmosphere too) whereas Io's atmosphere is whispy and not really much to write home about. The atoms/molecules/particles in the atmosphere reflect light in all directions, allowing it to diffuse through a bit, giving us that gradient from light to dark. Io, on the other hand, only reflects light off its surface, leading to the solid terminator you can see in that image. Just something to think about when writing those realistic descriptive passages. ;-)

Update: I photoshopped some Jupiters into skies to give a size comparison with the full moon. You can see them here.