I came across this article when I was browsing arXiv: astro-ph and it was pretty cool, so I thought I'd share it with you all.
Now, the actual paper is pretty technical so I wouldn't bother reading it unless you're really keen. In it the author, Kipping, simulates a planet-moon system transiting its primary and discusses the possibility of actually detecting the moon from such a transit.
The transit method, if you recall from last Wednesday's post, uses the fact that a planet passing in front of its sun dims the star's light a little bit. Kepler is a telescope orbiting in space which is currently searching for planets using this method.
In his paper, Kipping computes what we need to look for in the light curves (the data which shows how much light we can see from a star over time) to identify possible extra solar moons. What really caught my eye, though was that he concludes that Kepler is sensitive enough to detect some exo-moons. How cool is that? Maybe in the near future we'll be reading about the latest batch of exo-moons as well as the increasingly large database of exoplanets we're building up.
Monday, June 13, 2011
Wednesday, June 8, 2011
Planet spotting
I have in the past talked about some of the things you need to consider when you make up your own planets outside of the solar system. This week, I thought I'd talk about real exoplanets that we've discovered and how those discoveries have happened.
Methods of detection
There are several different ways in which we can determine whether a star has planet orbiting it.
Direct Imaging
This is sort of what it sounds like. Point a telescope and fortuitously see the planet. The problem is, most planets are quite small and not very bright, so this isn't the most reliable of methods. That's not to say it hasn't had some results. For example, Formalhaut B was discovered this way when the debris cloud surrounding the star was imaged.
Spectroscopically
This method requires a little bit more background physics. The Doppler effect is what happens when something which is emitting waves (for example, sound waves) is moving with regards to the observer. For example, if you are on a train, going past a level crossing with the ding-ding-ding-ing, it sounds higher-pitched when you're moving towards it because the waves seem more "bunched up", and then, as you go past, it suddenly sounds lower-pitched because they seem more "spread out". That's the Doppler effect.
Light is also a wave (or at least, often behaves as one). When a light source is moving towards you, the light waves will appear more bunched up and hence bluer (because blue light is a higher frequency than other visible colours). And if the light source, a star, for example, is moving away from you, the light will seem more spread out and hence redder.
Planets have a non-zero mass which means that while their star gravitationally pulls on them and keeps them in orbit, the planet also pulls on the star a bit. But because the planet is going around the star, it pulls in different directions at different times, making the star wobble. When the star wobbles towards Earth, it's light will look slightly bluer, and when it wobbles away, it will look slightly redder. Fancy spectrographs on telescopes can detect these slight variations in light output and hence, we can use this method to detect plants.
This only works on sufficiently heavy planets, sufficiently close to their stars. This has been the most popular method (up until Kepler, maybe, which I'll talk about below) for discovering exoplanets. It was the reason we suddenly discovered a whole lot of "hot Jupiters"—Jupiter sized, or bigger, planets close in to their stars—and had our pre-existing theories of planetary formation turned on their heads*.
*We were basing our theories on our own solar system which has large planets far out and small rocky planets close in. Suddenly, because of the detection methods we were capable of, we were seeing a lot of large planets close to their suns which we could not easily explain. However, it's quite likely that the plentitude of these hot Jupiters is actually a selection bias (as they are the easiest to find) rather than an indication that they are actually proportionally that common in the galaxy.
Transit
This method uses the fact that when a planet passes in front of its sun, it blocks out some (very small amount) of its light. Sensitive telescopes can pick up this dip in light output. The size of the dip gives an indication of the size of the planet and monitoring the star for long enough allows us to work out how quickly it goes around the star.
This is the method the Kepler telescope, currently in orbit, is using the discover a pile of exoplanets. And I do mean pile. So far more than 1200 planet candidates have been detected by Kepler. There are some really interesting stranger-than-fiction ones that I'll talk about in a later blog post.
Microlensing
Gravitational microlensing is a smaller-scale version of gravitational lensing, which I have briefly mentioned in the past. Remember (or follow the previous links to find out), if a massive object passes in front of a more distant light-source, then the massive object's gravity, which distorts spacetime a little bit, will cause the light from the background source to bend through the distortion. On a large scale, this can give us several images of the background source (example: Einstein's Cross). On a smaller scale (one might say a micro scale, heh), what happens when a moderately massive object like a star or a planet (or a star with a planet) passes in front of a background star, is the light from the background star is temporarily magnified.
If the foreground object is a star with a (sufficiently massive) planet around it, then the presence of the planet will make an extra peak in the magnification light curve. The height of this peak tells us about the mass and distance from the star of the planet.
Pulsar timing
I've put this in for completion, but there is no way for a pulsar to be human-habitable, even though some of them have planets. Pulsars are neutron stars—very, very dense stars made entirely of neutrons; sort of giant atoms. they emit radiation, mostly radio waves, from their magnetic poles which, because they spin very quickly, flashes past us in pulses, hence the name.
We can measure and time these pulses very precisely and usually vary in a very predictable way. Slight timing variations due to the gravitational tug of planets are very noticeable and some of the earliest exoplanets in the 90s were detected in this way.
Limitations
The problem with most of these methods is that they require a fortuitous positioning of planets with respect to their suns and the Earth. We cannot yet look at a star and definitely say that there are no planets orbiting it. If we're lucky we can say there there definitely are planets there, but if we don't see any it could be because they're not lined up with us nicely, rather than because they're not there.
Methods of detection
There are several different ways in which we can determine whether a star has planet orbiting it.
- Direct imaging
- Spectroscopically
- Transit
- Microlensing
- Pulsar timing
- Stellar wobble
Direct Imaging
This is sort of what it sounds like. Point a telescope and fortuitously see the planet. The problem is, most planets are quite small and not very bright, so this isn't the most reliable of methods. That's not to say it hasn't had some results. For example, Formalhaut B was discovered this way when the debris cloud surrounding the star was imaged.
Spectroscopically
This method requires a little bit more background physics. The Doppler effect is what happens when something which is emitting waves (for example, sound waves) is moving with regards to the observer. For example, if you are on a train, going past a level crossing with the ding-ding-ding-ing, it sounds higher-pitched when you're moving towards it because the waves seem more "bunched up", and then, as you go past, it suddenly sounds lower-pitched because they seem more "spread out". That's the Doppler effect.
Light is also a wave (or at least, often behaves as one). When a light source is moving towards you, the light waves will appear more bunched up and hence bluer (because blue light is a higher frequency than other visible colours). And if the light source, a star, for example, is moving away from you, the light will seem more spread out and hence redder.
Planets have a non-zero mass which means that while their star gravitationally pulls on them and keeps them in orbit, the planet also pulls on the star a bit. But because the planet is going around the star, it pulls in different directions at different times, making the star wobble. When the star wobbles towards Earth, it's light will look slightly bluer, and when it wobbles away, it will look slightly redder. Fancy spectrographs on telescopes can detect these slight variations in light output and hence, we can use this method to detect plants.
This only works on sufficiently heavy planets, sufficiently close to their stars. This has been the most popular method (up until Kepler, maybe, which I'll talk about below) for discovering exoplanets. It was the reason we suddenly discovered a whole lot of "hot Jupiters"—Jupiter sized, or bigger, planets close in to their stars—and had our pre-existing theories of planetary formation turned on their heads*.
*We were basing our theories on our own solar system which has large planets far out and small rocky planets close in. Suddenly, because of the detection methods we were capable of, we were seeing a lot of large planets close to their suns which we could not easily explain. However, it's quite likely that the plentitude of these hot Jupiters is actually a selection bias (as they are the easiest to find) rather than an indication that they are actually proportionally that common in the galaxy.
Transit
This method uses the fact that when a planet passes in front of its sun, it blocks out some (very small amount) of its light. Sensitive telescopes can pick up this dip in light output. The size of the dip gives an indication of the size of the planet and monitoring the star for long enough allows us to work out how quickly it goes around the star.
This is the method the Kepler telescope, currently in orbit, is using the discover a pile of exoplanets. And I do mean pile. So far more than 1200 planet candidates have been detected by Kepler. There are some really interesting stranger-than-fiction ones that I'll talk about in a later blog post.
Microlensing
Gravitational microlensing is a smaller-scale version of gravitational lensing, which I have briefly mentioned in the past. Remember (or follow the previous links to find out), if a massive object passes in front of a more distant light-source, then the massive object's gravity, which distorts spacetime a little bit, will cause the light from the background source to bend through the distortion. On a large scale, this can give us several images of the background source (example: Einstein's Cross). On a smaller scale (one might say a micro scale, heh), what happens when a moderately massive object like a star or a planet (or a star with a planet) passes in front of a background star, is the light from the background star is temporarily magnified.
If the foreground object is a star with a (sufficiently massive) planet around it, then the presence of the planet will make an extra peak in the magnification light curve. The height of this peak tells us about the mass and distance from the star of the planet.
Pulsar timing
I've put this in for completion, but there is no way for a pulsar to be human-habitable, even though some of them have planets. Pulsars are neutron stars—very, very dense stars made entirely of neutrons; sort of giant atoms. they emit radiation, mostly radio waves, from their magnetic poles which, because they spin very quickly, flashes past us in pulses, hence the name.
We can measure and time these pulses very precisely and usually vary in a very predictable way. Slight timing variations due to the gravitational tug of planets are very noticeable and some of the earliest exoplanets in the 90s were detected in this way.
Limitations
The problem with most of these methods is that they require a fortuitous positioning of planets with respect to their suns and the Earth. We cannot yet look at a star and definitely say that there are no planets orbiting it. If we're lucky we can say there there definitely are planets there, but if we don't see any it could be because they're not lined up with us nicely, rather than because they're not there.
Wednesday, June 1, 2011
Lifting life off Earth
A friend suggested I blog about getting a biosphere off Earth and onto another planet (within the solar system). This is a bit of a challenge for me since, if I am anything, then I am not a biologist. As a result, I am going to focus on the transportation logistics more than what, specifically, we would need to get to said other planet.
However, if you are interested in the what and how to maintain it once there, I strongly suggest reading these two posts by Patty Jansen: So you want to be a space farmer (part 1) and Growing crops in space (part 2). She has a background in agricultural science and hence is significantly more knowledgeable than I on the matter.
Getting off the ground
Lifting anything off Earth into orbit requires a large chunk of energy. Exactly how much depends mainly on the weight (and a little bit on the size, in the sense of how big—and hence heavy—the spaceship doing the lifting needs to be). To get something off Earth (pretty much to anywhere further away than the moon, though it's not that different for the moon either), we need to give it enough energy to overcome the energy of Earth's gravitational pull.
Some basic terminology first:
OK, so that's the basics. To get something off the ground and into space, we need to make it go pretty fast to overcome Earth's gravitational pull. However, we don't do it all in one go; it's just not logistically a brilliant idea. Among other things, the faster you go within Earth's atmosphere, the greater air resistance (the friction air exerts on the spaceship) is. This is why rockets usually have stages. The space shuttles, for instance, had two initial boosters to get it off the ground, another large rocket to get them out of the atmosphere, then some small rockets which stay attached to the shuttle (the others are discarded when the fuel is used up) for orbital manoeuvring and coming back down to Earth. Here is an infographic from Wiki.
The tricky thing, when we're talking about getting the elements of an ecosystem off the ground, is how much they weigh. I don't know of any reason why crops wouldn't be transported as seeds which, compared with plants weigh a lot less and take up a lot less room. Depending on where you're taking them, giving them enough water and the right kind of soil is likely to be much more of a problem. In most cases, I think it would be best to mine the necessary water from whatever nearby source you can (the rings of Saturn, mayhaps?) and possibly ditto with the minerals needed for soil, but see Patty's posts linked to up top because I'm far from an expert.
Animals, however, would be a lot harder. With our current technology, we can't really transport a bunch of animals in foetus form and then grow them when we get to wherever like we can with plant seeds. Animals weigh a lot and eat a lot and produce a lot of waste products. That sort of thing (while making good fertiliser for our off-world plants) would be very difficult to transport.
To put this in a bit of perspective, let's look at the weight and lifting capacity of the space shuttle (which have almost all been decommissioned now with only Atlantis having one mission left). According to Wiki, an empty space shuttle weighs close to 70 000 kg, has a maximum payload weight of 25 000 kg and the payload bay is 4.6 times 18.3 metres (doesn't say how tall, but let's assume tall enough for cows). Twenty-five thousand kilograms may seem like a lot, but remember that the shuttles have been used to lift several bits of International Space Station into orbit. So a cow weighs around 500 kg, depending on the type, but let's run with this number because it's round and convenient. That means theoretically, we could squish 500 cows into a space shuttle and lift them into orbit. Well, that's not very helpful because we're ignoring all the food and water they'd need. I also think they wouldn't quite actually fit into the payload bay. Let's say cows need a metre by two metres of space to stand around in. That leaves us with only around 70 cows in our cargo bay. Well, OK, that means we could use the rest of the space for that pesky food and water I keep mentioning...
Let's say a cow eats 50 kg of hay a day... WolframAlpha tells me that hay weighs around 380 kg for a cubic metre (when pressed, because anything else would be silly in this context...) Our 70 cows would eat more than nine cubic metres of pressed hay a day and since getting to anywhere other than the moon takes at least months... we quickly run into problems being able to carry enough, even without worrying about the water.
It's fair, at this point, to mention that the space shuttles were obviously not designed to carry cows anywhere. They were designed to carry bits of ISS and other space-based equipment into orbit and not further. But in terms of lifting power they and the Soyuz rockets are all we've currently got. Also, cows probably aren't the best thing to start off carrying to other planets, I was just trying to make a point.
It takes a lot of energy to launch anything into space, let alone an ecosystem. Unlike the ISS which was built by launching bits up in manageable chunks, it's not really practical to do that with live animals. Flora poses less of a challenge and the difficult part becomes setting it up sensibly on the other end. Also the part where we haven't actually sent people on very long space flights yet.
More on this topic at a later date.
However, if you are interested in the what and how to maintain it once there, I strongly suggest reading these two posts by Patty Jansen: So you want to be a space farmer (part 1) and Growing crops in space (part 2). She has a background in agricultural science and hence is significantly more knowledgeable than I on the matter.
Getting off the ground
Lifting anything off Earth into orbit requires a large chunk of energy. Exactly how much depends mainly on the weight (and a little bit on the size, in the sense of how big—and hence heavy—the spaceship doing the lifting needs to be). To get something off Earth (pretty much to anywhere further away than the moon, though it's not that different for the moon either), we need to give it enough energy to overcome the energy of Earth's gravitational pull.
Some basic terminology first:
- Kinetic energy is the energy something has due to its movement. It mostly depends on how fast the object is moving, but also on its mass. A faster object will have more kinetic energy, but of two objects moving at the same speed, the heavier one will have more kinetic energy. The formula for kinetic energy is
- Potential energy is stored energy that has the potential to turn into a more directly useful form of energy. For example, if you lift a brick off the ground, you're giving it the potential to turn gain kinetic energy when you drop it. In fact, thanks to conservation of energy, the amount of potential energy you add to it when you lift it up will be equal to the kinetic energy it gains as it falls and just before it hits the ground. What we are interested in is gravitational potential energy. You can have other types, like spring potential energy, which is the energy stored in a spring when it is stretched or compressed. The general formula for gravitational potential energy is
- Conservation of energy is the law that says energy cannot be created or destroyed but can only change forms. Hence, potential energy can be transformed into kinetic energy and vice versa, but neither can appear out of nothing. (On a macroscopic level. Things get a little bit more complicated on a quantum scale, but that's not relevant here.)
- Escape velocity is the velocity required to get something off the surface of a planet. It's basically the amount of kinetic energy required to overcome the potential energy stored between your object and the planet. It's found by equating the kinetic and potential energies above (ignore the minus sign, it's just a convention). Doing that, the mass for the object, m, cancels out and we find a common escape velocity depending only on the mass of the planet, M, and the radius of the planet, R. (This is assuming we're trying to get off the surface. For other distances, replace radius with distance from the centre of the planet.) Incidentally, Earth's escape velocity is about 11 kilometres per second (more than 40 000 km/h). The general formula for escape velocity is
OK, so that's the basics. To get something off the ground and into space, we need to make it go pretty fast to overcome Earth's gravitational pull. However, we don't do it all in one go; it's just not logistically a brilliant idea. Among other things, the faster you go within Earth's atmosphere, the greater air resistance (the friction air exerts on the spaceship) is. This is why rockets usually have stages. The space shuttles, for instance, had two initial boosters to get it off the ground, another large rocket to get them out of the atmosphere, then some small rockets which stay attached to the shuttle (the others are discarded when the fuel is used up) for orbital manoeuvring and coming back down to Earth. Here is an infographic from Wiki.
The tricky thing, when we're talking about getting the elements of an ecosystem off the ground, is how much they weigh. I don't know of any reason why crops wouldn't be transported as seeds which, compared with plants weigh a lot less and take up a lot less room. Depending on where you're taking them, giving them enough water and the right kind of soil is likely to be much more of a problem. In most cases, I think it would be best to mine the necessary water from whatever nearby source you can (the rings of Saturn, mayhaps?) and possibly ditto with the minerals needed for soil, but see Patty's posts linked to up top because I'm far from an expert.
Animals, however, would be a lot harder. With our current technology, we can't really transport a bunch of animals in foetus form and then grow them when we get to wherever like we can with plant seeds. Animals weigh a lot and eat a lot and produce a lot of waste products. That sort of thing (while making good fertiliser for our off-world plants) would be very difficult to transport.
To put this in a bit of perspective, let's look at the weight and lifting capacity of the space shuttle (which have almost all been decommissioned now with only Atlantis having one mission left). According to Wiki, an empty space shuttle weighs close to 70 000 kg, has a maximum payload weight of 25 000 kg and the payload bay is 4.6 times 18.3 metres (doesn't say how tall, but let's assume tall enough for cows). Twenty-five thousand kilograms may seem like a lot, but remember that the shuttles have been used to lift several bits of International Space Station into orbit. So a cow weighs around 500 kg, depending on the type, but let's run with this number because it's round and convenient. That means theoretically, we could squish 500 cows into a space shuttle and lift them into orbit. Well, that's not very helpful because we're ignoring all the food and water they'd need. I also think they wouldn't quite actually fit into the payload bay. Let's say cows need a metre by two metres of space to stand around in. That leaves us with only around 70 cows in our cargo bay. Well, OK, that means we could use the rest of the space for that pesky food and water I keep mentioning...
Let's say a cow eats 50 kg of hay a day... WolframAlpha tells me that hay weighs around 380 kg for a cubic metre (when pressed, because anything else would be silly in this context...) Our 70 cows would eat more than nine cubic metres of pressed hay a day and since getting to anywhere other than the moon takes at least months... we quickly run into problems being able to carry enough, even without worrying about the water.
It's fair, at this point, to mention that the space shuttles were obviously not designed to carry cows anywhere. They were designed to carry bits of ISS and other space-based equipment into orbit and not further. But in terms of lifting power they and the Soyuz rockets are all we've currently got. Also, cows probably aren't the best thing to start off carrying to other planets, I was just trying to make a point.
It takes a lot of energy to launch anything into space, let alone an ecosystem. Unlike the ISS which was built by launching bits up in manageable chunks, it's not really practical to do that with live animals. Flora poses less of a challenge and the difficult part becomes setting it up sensibly on the other end. Also the part where we haven't actually sent people on very long space flights yet.
More on this topic at a later date.
Friday, May 27, 2011
Foreign Skies: Daytime
When I wrote my first post on this blog, I wanted to include some photoshopped images of to-scale Jupiter hanging in the sky as it would above Ganymede and Io. At the time, I didn't have any good photos of the moon to compare with and paste over and I didn't want to steal something from Google Image Search so I put it off. I still don't have any good, cloudless photos of the moon at night with a suitably urban back drop. Instead, I decided to put my (very average) photoshopping skills to use and make a daytime Jupiter in the sky.
A caveat: Ganymede and Io both lack atmospheres that even remotely resemble Earth's. As a result, you wouldn't get a blue sky on either, even under a biodome of some sort (there wouldn't be enough air in the dome to have that effect). Instead the sky would be black outside of the giant orb of Jupiter (which, when full, would definitely be bright enough to make it hard to see stars).
So you shouldn't treat these images as "what the sky would look like on Ganymede/Io" more as an indication of how large Jupiter would be in the Earth's sky if you put it in Ganymede's/Io's place.
First! The original photo with the Earth's natural moon left in (taken with my phone, so it could be awesomer, but it served its purpose):
 |
| Unlike Earth, Ganymede and Io are unlikely ever have gum trees. Just saying. (Click to enlarge.) |
For the remaining compositions, this APOD image is the shot of Jupiter I used. Credit to NASA, ESA, and E. Karkoschka (U. Arizona).
Next up, Jupiter as seen from Ganymede. I left the moon in for the first one, just so you can visually compare sizes better:
 | ||
| To scale, albeit physically unrealistic. I don't think Jupiter would quite be that colour either (particularly the edges wouldn't be so dark, but I couldn't fix it) and of course the lighting is all wrong. (Click to enlarge.) |
Just Jupiter alone in the Ganymedian sky:
 |
| Jupiter in Earth's sky if Earth was in Ganymede's orbit around the gas giant. (Click to enlarge.) |
And, last one, the size of Jupiter in Io's sky. Large doesn't really cover it.
 |
| Jupiter looming as though over the Io skyline. Of course, Io is even less likely than Ganymede to have apartment buildings on it, but shh! (Click to enlarge.) |
And there you have it. Once I have a decent night skyline with the moon in it, I will repeat this but at night. It will be significantly awesomer. I just have to get some night photography in first.
EDIT: I have made some similar images for Jupiter from Europa and Callisto and the sun as seen from the Jovian system. See my new post here.
Wednesday, May 25, 2011
Conquering the Horizon
And now for something completely different. The horizon; how far away is it? How different would it look on another planet? We're used to horizons on Earth but if we're writing a story set on an asteroid or on a small moon or planet the horizon will be closer because the planet's surface falls away more quickly.
Distance to the Horizon
When I talk about the distance to the horizon, what I mean is if you're somewhere flat, what's the furthest you can see (including with the aid of binoculars or a telescope etc)? On Earth a good example of this would be how far away the point where the sea meets the sky is, when you're standing on a jetty. Once we have trees and buildings in the way it can get a bit more complicated. Luckily, the sort of extraterrestrial locales where the horizon is going to be most different to Earth's are least likely to have a large abundance of trees. Convenient.
Working out how far away the horizon is takes a little bit of trigonometry. I've drawn a sketch below of all the relevant distances and whatnot.
The important thing to that you need to know about your non-Earth planet is how big it is or, more specifically, it's radius which is labelled R in the image above. It's a reasonable assumption that you'll have at least a rough idea of how tall your characters are. If you don't, it doesn't really matter, you can just guess something close since there's not going to be much difference between a tall person's horizon and a short person's (the differences really come into play in non-horizon situations, such as crowds). For the purposes of my calculations later on, I'm going to set h = 1.7 meters. Because I can.
Now, that's a right angle between the line I've labelled d and the left hand radius line. Since we know R and h we can now use Pythagoras's Theorem to work out the distance d. Don't worry if you don't remember any maths, I'm just going to tell you the answer.
Chances are, your planet is significantly larger than your person, so you will usually be able to ignore the h2 but not always (if on a small asteroid, for example). If in doubt, leave it in. It won't make your answer worse.
This is not an unhelpful result. However, I can't help but feel that when people stand in a tall tower and say things like "They're ten miles away but gaining ground!" they don't mean ten miles from their eyes, but ten miles from the bottom of the tower (if nothing else, they'd probably be estimating based on land marks and those are definitely relative to the ground distance).
So how do we find the distance along the ground, s? Unsurprisingly, with more maths. We use the fact that s = Rθ and then work out θ so we can substitute for it and not have to actually calculate it directly. Using the same triangle as before, we can find s in two different ways:
For planets/moons which are much, much larger than a person, s and d will be very close; it's the tiny, tricksy moons or asteroids are where it'll really make a difference.
So how far?
Some examples now for a person 170 cm tall and for a ten storey building (30 metres high):
Seeing things beyond the horizon
The horizons I've talked about above are the limiting distances for seeing things on (or close to) the ground. Things are a little bit different if we want to work out from how far away we can start to see the top of a tall building, for example.
The diagram below shows that although my little stick figure can only see the ground up to d distance away, s/he can see the top of a building which is d+b distance away. Huzzah!
So what if we want to work out how from how far away we start to see the top of a building/monument/spaceport/volcano? Easy. All we have to do is work out the distances to the horizon for both the person and the building/monument/spaceport/volcano and add them together. The distance, D, from which the person starts to see the top of the building/whatever is then approximately given by:
You may have noticed that if we want to work out the distance from which someone can start to see a ten storey building, all we have to do is add the horizons I worked out above together. How convenient! Just quickly, the distances at which the building will start looming out of the ground are:
Distance to the Horizon
When I talk about the distance to the horizon, what I mean is if you're somewhere flat, what's the furthest you can see (including with the aid of binoculars or a telescope etc)? On Earth a good example of this would be how far away the point where the sea meets the sky is, when you're standing on a jetty. Once we have trees and buildings in the way it can get a bit more complicated. Luckily, the sort of extraterrestrial locales where the horizon is going to be most different to Earth's are least likely to have a large abundance of trees. Convenient.
Working out how far away the horizon is takes a little bit of trigonometry. I've drawn a sketch below of all the relevant distances and whatnot.
 |
| R is the radius of the planet/moon, h is the height of your person (well, of their eyes) or if they're in a building, it can be how high up they are, d is the straight line distance to the horizon, s is the distance along the surface of the planet/moon and θ is an angle that will be useful in some calculations. |
Now, that's a right angle between the line I've labelled d and the left hand radius line. Since we know R and h we can now use Pythagoras's Theorem to work out the distance d. Don't worry if you don't remember any maths, I'm just going to tell you the answer.
Chances are, your planet is significantly larger than your person, so you will usually be able to ignore the h2 but not always (if on a small asteroid, for example). If in doubt, leave it in. It won't make your answer worse.
This is not an unhelpful result. However, I can't help but feel that when people stand in a tall tower and say things like "They're ten miles away but gaining ground!" they don't mean ten miles from their eyes, but ten miles from the bottom of the tower (if nothing else, they'd probably be estimating based on land marks and those are definitely relative to the ground distance).
So how do we find the distance along the ground, s? Unsurprisingly, with more maths. We use the fact that s = Rθ and then work out θ so we can substitute for it and not have to actually calculate it directly. Using the same triangle as before, we can find s in two different ways:
 |
| If you're wondering, cos and tan are trigonometric functions all scientific calculators (including the ones hiding in all your computers) can do. The -1 indicates that's it's the inverse of the function which you can usually access by pressing shift/2nd or something like that, depending on the calculator. |
For planets/moons which are much, much larger than a person, s and d will be very close; it's the tiny, tricksy moons or asteroids are where it'll really make a difference.
So how far?
Some examples now for a person 170 cm tall and for a ten storey building (30 metres high):
- On Earth, ignoring atmospheric effects which actually extend the apparent horizon thanks to bending light, the horizon is 4.7 km away. From a ten storey building it's 19.7 km.
- On the moon or Io, which are similar in size, a standing horizon is just under 2.5 km and the ten storey building horizon is about 10 km.
- Ganymede and Titan (moons of Jupiter and Saturn, respectively) are a bit bigger than those two, with standing and ten storey building horizons of 3 km and 12.5 km.
- Mars is about one and a half times the size of Ganymede and a bit more than half the size of Earth. It has horizons 3.3 km and 14 km for standing and building respectively.
- Deimos, Mars's moon, is rounder than Mars's other moon, Phobos, but still not that round. If you stand on a fortuitously round bit, the horizon will be 140 meters away (that's right, metres not kilometres—Deimos is actually an oblong with dimensions only 15⨉12.2⨉10.4 km across. Its average radius is 6.2 km). If you somehow managed to put a 10 storey building on it... well you'd see about 600 meters away.
- Ceres is a large, round asteroid (or dwarf planet) in the asteroid belt. It was one of the bodies that, when Pluto's planethood was called into question, was up for being classified as a planet if Pluto got to stay. (If you're wondering, it is considerably larger than Deimos, with a radius of 471 km.) A person would see the horizon 1.3 km away (pretty close if you think about how far a kilometre looks when you're driving, for example) and a ten storey building would see the horizon drop off 5.3 km away.
- And speaking of Pluto, Pluto's largest moon (it also has two tiny ones), Charon, is a bit bigger than Ceres and has a standing horizon of 1.4 km and a building horizon of 6 km.
Seeing things beyond the horizon
The horizons I've talked about above are the limiting distances for seeing things on (or close to) the ground. Things are a little bit different if we want to work out from how far away we can start to see the top of a tall building, for example.
The diagram below shows that although my little stick figure can only see the ground up to d distance away, s/he can see the top of a building which is d+b distance away. Huzzah!
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You may have noticed that if we want to work out the distance from which someone can start to see a ten storey building, all we have to do is add the horizons I worked out above together. How convenient! Just quickly, the distances at which the building will start looming out of the ground are:
- Earth: 25.5 km (assuming you can find an isolated ten storey building in the middle of a 25 km circle of flat ground...)
- Moon/Io: 12.5 km
- Ganymede/Titan: 15.5 km
- Mars: 17 km
- Deimos: 740 meters if you can manage an ideal situation (I strongly suspect that you can't, but these calculations do give you a good idea of how small Deimos is... the edge of Deimos would look so close!)
- Ceres: 6.6 km
- Charon: 7.4 km
Monday, May 23, 2011
A couple of cool Io animations
So I was browsing around NASA's photo archive looking for a nice high-res image of Io and I came across this little movie of Jupiter-shine on Io. It's a time-lapse movie in which you can see the Io moving so that the sun is on its far side — it starts off behind and to the left which is why you can see a bright crescent at first. As the sun moves behind Io from the camera's point of view, Jupiter, which is behind the camera becomes more fully illuminated. The sunlight bouncing off Jupiter in turn more brightly illuminates Io's surface which is why the night side of Io gets darker as the crescent of Io gets smaller. Pretty cool, eh?
You can read more about planetshine in this post, which also talks about how much light you'd get from the sun at different distances.
Another cool little movie on the same site is this one, which shows some interesting happenings on Io's dark side. The bright spots away from the edges are volcanoes, whereas the two blue glows on the edges are similar to aurorae on Earth. Except those aren't the poles, but rather at the equator where charged particles are colliding with the tenuous wisps of Io's atmosphere. The thin atmosphere itself is only existent because it's constantly being replenished by volcanic gases. Nifty.
You can read more about planetshine in this post, which also talks about how much light you'd get from the sun at different distances.
Another cool little movie on the same site is this one, which shows some interesting happenings on Io's dark side. The bright spots away from the edges are volcanoes, whereas the two blue glows on the edges are similar to aurorae on Earth. Except those aren't the poles, but rather at the equator where charged particles are colliding with the tenuous wisps of Io's atmosphere. The thin atmosphere itself is only existent because it's constantly being replenished by volcanic gases. Nifty.
Wednesday, May 18, 2011
Ringing Tides
Saturn has rings. So do all the other gas giants in the solar system. Although we have no ability to confirm whether extra solar gas giants also have rings, chances are some do. Where do these rings come from? Why doesn't Earth have any?
Terrible Tides
The answer to the first question, as you may have guessed from the title of this post, is tides. Last week I talked about tides causing satellites to be locked in synchronous orbits around their primaries (recommended reading if you haven't already). The thing to remember now is that the side of a satellite closest to its primary experiences a stronger gravitational pull than the far side. The difference in forces depends on the mass of the primary, the distance of the satellite from the primary and the size if the satellite.
If you recall from the introduction to gravity post, the force of gravity exerted on a mass, m, a distance, r, from another mass, M, is given by:
If we take M to be the mass of the primary and then consider two smaller masses m1 and m2, one of which is located at r1 on the near side of the satellite, and the other at r2, the far side of the satellite. If we assume our two small masses are equal (you can think of it as considering a kilogram of moon rock in two different locations), then the ratio of the forces experienced by them will be:
That equation might seem a bit abstract, so let's look at it in the context of a few real examples.
It is now possible to come up with a scenario where the pull of the primary on the near side of the satellite is bigger than the pull of its own gravity. Let's look at one of Saturn's tiny moonlets. Pan orbits inside Saturn's A ring (towards the outer edge of the ring system). It's radius is only 14.2 km and it weighs 5⨉1015 kg, making it's surface acceleration due to gravity 0.0016 m/s2 (less than a ten thousandth of a percent that of Earth's). By comparison, the acceleration due to gravity from Saturn at that distance is 2.12 m/s2, more than 1300 times greater. Clearly, Pan could not have formed where it now orbits since it's very much held together by chemical forces, not gravitational.
EDIT: Correction made to the above paragraph. Previously I had stated that if you stood on the Saturn-side of Pan you would fall up into Saturn. This is not true. The more accurate statement I should've made was that if you were floating around in the vicinity of Pan's orbit and Pan came past you, its gravity would not be strong enough to pull you in over Saturn's gravity. No matter how close to it you were (even if you could touch the surface), if you were not already moving along with it (and hence had enough centripetal acceleration to balance Saturn's gravitational acceleration), then Saturn's gravity would win out and you would fall towards Saturn, rather than towards Pan.
Making rings
Even further out than the point at which the primary's gravity becomes stronger than the satellite's gravity, the primary's gravity will start to deform the satellite. This effect is not dissimilar to the tidal bulge the moon causes on Earth. It is also part of the reason the Galilean moons of Jupiter are tidally locked.
(Interesting fact: over time, the tidal bulge of the Earth is causing the Earth to slow down its period of rotation since the change in shape (which isn't constant, remember, as the moon's orbit is much slower than the Earth's day) alters the way it rotates (catch phrase: conservation of angular momentum). The drag of the water in the tidal bulge is also pushing the moon back, slightly, in its orbit. Eventually (and we're talking a pretty long eventually) the Earth-moon system will settle into a mutually tidally locked rotation with the moon significantly further away than it is now. Here is an interesting article about it from Space.com.)
In the case of a satellite which is reasonably fluid and only being held together by its own gravitational pull (called self-gravity), then there is no reason for it to stay together as one lump. It will disintegrate because the part closer to the primary wants to orbit faster than the part further away. A disintegrated satellite will turn into a system of rings around the planet. The point at which this happens is called the Roche limit and the equation which tells us the distance from the primary of the Roche limit is:
You'll notice that there are densities in the above formula. The density of the satellite is relevant because it's a measure of both mass and gravity (since we're talking satellites that are only held together by their self gravity and not chemical bonds). The density of the primary comes into it because we need to know the mass (which is proportional to radius cubed times density and R will becomes cubed if you move it inside the brackets) but the radius is also relevant because if the Roche limit is inside the primary, we can pretty much ignore it.
Of course, most satellite aren't balls of dust but are held together by other chemical forces (like Pan is). For example a rock on Earth isn't held together by gravity, it's held together by the chemical bonds between the different atoms and molecules inside (slightly different bonds depending on it's composition). Similarly, once a satellite has formed (outside of the Roche limit), then it probably goes through other experiences (such as tidal heating) which fuse it into a more solid lump. If it then wanders inside the Roche limit, it's not going to dissolve just because it couldn't've formed there. Pan and a handful of other moons in Saturn's rings prove this point. So what's the Roche limit for satellites held together by more than just gravitational forces? It sort of depends on the forces, but Roche himself derived an approximation for fluid satellites which deform a bit before they break up due to the tidal forces:
If you're wondering whether rock counts as fluid, it does. Everything will deform a bit under sufficiently strong forces.
So this last equation is the point at which a satellite will start to break up if it spirals in too close to its primary. For Earth-moon system, the moon will disintegrate if it wanders within 11 000 km. Luckily for the moon, this isn't likely to happen until the sun end's it's main sequence life.
Terrible Tides
The answer to the first question, as you may have guessed from the title of this post, is tides. Last week I talked about tides causing satellites to be locked in synchronous orbits around their primaries (recommended reading if you haven't already). The thing to remember now is that the side of a satellite closest to its primary experiences a stronger gravitational pull than the far side. The difference in forces depends on the mass of the primary, the distance of the satellite from the primary and the size if the satellite.
If you recall from the introduction to gravity post, the force of gravity exerted on a mass, m, a distance, r, from another mass, M, is given by:
If we take M to be the mass of the primary and then consider two smaller masses m1 and m2, one of which is located at r1 on the near side of the satellite, and the other at r2, the far side of the satellite. If we assume our two small masses are equal (you can think of it as considering a kilogram of moon rock in two different locations), then the ratio of the forces experienced by them will be:
That equation might seem a bit abstract, so let's look at it in the context of a few real examples.
- Io is 4.217⨉108 m from Jupiter, on average, and has a radius of 1.8⨉106 m. The pull of Jupiter's gravity on the far side is just 99.15% that on the near side.
- Doing a similar calculation for the moon (orbiting the Earth), we find far side gravity 99.10% that of near side.
- Mercury orbiting the sun has far side gravity 99.99% that of the near side since, even though it's very close to the sun, it's a lot further away than the moons are from their primaries.
- Let's look at Saturn now. Not one of Saturn's moons, but Saturn's rings. The main rings, according to Wiki, extend between 66 900 km and 480 000 km above the centre of Saturn. The gravitational pull from Saturn on the far edge is 83.6% that of the near edge. Compared with the solid bodies discussed above, that's a much more significant difference.
It is now possible to come up with a scenario where the pull of the primary on the near side of the satellite is bigger than the pull of its own gravity. Let's look at one of Saturn's tiny moonlets. Pan orbits inside Saturn's A ring (towards the outer edge of the ring system). It's radius is only 14.2 km and it weighs 5⨉1015 kg, making it's surface acceleration due to gravity 0.0016 m/s2 (less than a ten thousandth of a percent that of Earth's). By comparison, the acceleration due to gravity from Saturn at that distance is 2.12 m/s2, more than 1300 times greater. Clearly, Pan could not have formed where it now orbits since it's very much held together by chemical forces, not gravitational.
EDIT: Correction made to the above paragraph. Previously I had stated that if you stood on the Saturn-side of Pan you would fall up into Saturn. This is not true. The more accurate statement I should've made was that if you were floating around in the vicinity of Pan's orbit and Pan came past you, its gravity would not be strong enough to pull you in over Saturn's gravity. No matter how close to it you were (even if you could touch the surface), if you were not already moving along with it (and hence had enough centripetal acceleration to balance Saturn's gravitational acceleration), then Saturn's gravity would win out and you would fall towards Saturn, rather than towards Pan.
Making rings
Even further out than the point at which the primary's gravity becomes stronger than the satellite's gravity, the primary's gravity will start to deform the satellite. This effect is not dissimilar to the tidal bulge the moon causes on Earth. It is also part of the reason the Galilean moons of Jupiter are tidally locked.
(Interesting fact: over time, the tidal bulge of the Earth is causing the Earth to slow down its period of rotation since the change in shape (which isn't constant, remember, as the moon's orbit is much slower than the Earth's day) alters the way it rotates (catch phrase: conservation of angular momentum). The drag of the water in the tidal bulge is also pushing the moon back, slightly, in its orbit. Eventually (and we're talking a pretty long eventually) the Earth-moon system will settle into a mutually tidally locked rotation with the moon significantly further away than it is now. Here is an interesting article about it from Space.com.)
In the case of a satellite which is reasonably fluid and only being held together by its own gravitational pull (called self-gravity), then there is no reason for it to stay together as one lump. It will disintegrate because the part closer to the primary wants to orbit faster than the part further away. A disintegrated satellite will turn into a system of rings around the planet. The point at which this happens is called the Roche limit and the equation which tells us the distance from the primary of the Roche limit is:
 |
| d is the distance of the Roche limit from the centre of the primary, R is the radius of the primary and ⍴M and ⍴m are the densities of the primary and the satellite respecively. |
You'll notice that there are densities in the above formula. The density of the satellite is relevant because it's a measure of both mass and gravity (since we're talking satellites that are only held together by their self gravity and not chemical bonds). The density of the primary comes into it because we need to know the mass (which is proportional to radius cubed times density and R will becomes cubed if you move it inside the brackets) but the radius is also relevant because if the Roche limit is inside the primary, we can pretty much ignore it.
Of course, most satellite aren't balls of dust but are held together by other chemical forces (like Pan is). For example a rock on Earth isn't held together by gravity, it's held together by the chemical bonds between the different atoms and molecules inside (slightly different bonds depending on it's composition). Similarly, once a satellite has formed (outside of the Roche limit), then it probably goes through other experiences (such as tidal heating) which fuse it into a more solid lump. If it then wanders inside the Roche limit, it's not going to dissolve just because it couldn't've formed there. Pan and a handful of other moons in Saturn's rings prove this point. So what's the Roche limit for satellites held together by more than just gravitational forces? It sort of depends on the forces, but Roche himself derived an approximation for fluid satellites which deform a bit before they break up due to the tidal forces:
If you're wondering whether rock counts as fluid, it does. Everything will deform a bit under sufficiently strong forces.
So this last equation is the point at which a satellite will start to break up if it spirals in too close to its primary. For Earth-moon system, the moon will disintegrate if it wanders within 11 000 km. Luckily for the moon, this isn't likely to happen until the sun end's it's main sequence life.
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