Weird Worlds: Kepler 22b

It's what all the cool kids are (still) talking about, so why not a post on it? In the glut of Kepler mission transiting planets, a new planet has risen to the fore as the possibly most Earthlike extrasolar planet. Huzzah! (And in a few months yet another new planet will take its place. Just watch).

Credit: NASA/Ames/JPL-Caltech

What we know

Kepler 22 is a star very similar to our sun. It is slightly smaller (0.98 times the radius of the sun) and slightly smaller (0.97 times the mass of the sun). It is 587.1 light years away, so it's not something we're going to be visiting soon.

So far, it only has one known planet, dubbed -- as these things go -- Kepler 22b. This planet is smack bang in the habitable zone ...on the inner edge, so OK, maybe not that smack bang... BUT. The part people are getting exciting about is that it's the smallest planet we've found in the habitable zone thus far.

How small? 2.38 Earth radii. For comparison, Neptune, the next biggest solar planet after Earth, is 3.88 times Earth's radius. That puts Kepler 22b just on the smaller side of halfway between Earth and Neptune. In terms of size.

What we don't know is how massive it is. The way the Kepler satellite detects exoplanets is by looking at stars and measuring how much dimmer they get when (if) a planet passes between the star and us. By knowing how much light the star gives out usually, and how far away it is, it's possible to work out how large the planet is. (And, actually, even if you don't know how far away the star is, you could still work out how big the planet has to be as a fraction of the size of the star.)

We know how long its year is, basically by measuring how often it passes in front of the star. Yeah. It's that simple (particularly since there seems to be just one planet in this system at the moment). So Kepler 22b's year is 290 days long, not drastically different to Earth's. That and its position in the habitable zone have lead people to suppose that maybe it's not so different to Earth.

Based on how far it is from its star etc, the average surface temperature would be about -11º Celsius if it had no atmosphere. However, if it had an atmosphere like Earth's average temperature would be about 22º C. Perhaps a slightly less insulating atmosphere would be in order, however, since Earth's mean surface temperature is actually 14º C. (Because we have to take global lows and highs into account, you see, and average over the whole planet, never mind that 22º is just about room temperature.)

Harping on what we don't know

It's that mass. And the fact that we don't have a planet of that size in our solar system. It's in the transitional zone, as I said, between rocky terrestrial planet and mini gas giant. If it's rocky, it's likely that the gravity will be stronger than Earth's, thanks to it's larger size. In fact, if it's the same density as Earth, the surface gravity will be about 2.4 times Earth. Which isn't long-term viable. At Neptune's density, we get a ball of gas with surface gravity 0.72 times Earth. A ball of ice would result in gravity 4.3 times Earth. The least dense terrestrial planet in our solar system is Mars and at Mars' density, Kepler 22b would have gravity 1.7 Earth's.

So there's quite a spectrum of possibilities there. We really don't know enough to make any definitive judgements at this stage. That doesn't make it less exciting, of course, and it's true that there might be oceans and continents there. Or it could be a small dense ball with a thick gaseous atmosphere inhabited by dragons (perhaps even fire-breathing ones, if there's methane in the atmosphere and breathing fire is a viable way to catch prey). Which would also be cool. Arguably, cooler.

Weird Worlds: Kepler 14b

This post follows the same theme as last week's: planets in binary star systems. Last week's planet, Kepler 16(AB)b, orbited two stars which closely spun around each other. This week's planet, Kepler 14b, orbits one star, which in turn orbits another star at a further distance. This paper, by Buchhave et al (2011), is the main source of my data on Kepler 14b.

Kepler 14b is unlikely to have any sort of solid surface.
However, the two stars could look something like this. Maybe.
Honestly, I think this might be a re-purposed Io illustration
but NASA did use it on a Kepler 14b page, so who am I to argue?
Image credit: Susan Stanley for NASA Kepler Mission Education
and Public Outreach.

It turns out that unlike originally thought from the Kepler satellite data, the star Kepler 14 is not a star, but is two stars. Buchhave et al (2011) discovered this when they went and looked at it with an Earth-based optical telescope. The two stars are so close together in the sky as seen from Earth that it took fancy adaptive optics to be able to resolve them. Resolving, by the way, means being able to distinguish that there are two separate sources of light, not a single blob. The stars were too close together to be able to separate out their spectra (because this uses a different instrument than the one that images them normally), so a few assumptions had to be made, but none that should strongly affect the planet.

Some facts

The two stars that make up Kepler 14 are designated A and B with A being the one the planet orbits, but the nomenclature is a little fuzzy because the binary nature of the system was discovered later. Going on the basis that the system is 980 parsecs away (3200 light years), then the two stars are separated by 280 AU (remember, 1 AU is the distance from Earth to sun and Pluto is about 40 AU from the sun). Both stars are bigger, hotter and brighter than the sun. The sun is a G type star and the Kepler 14 stars are F types, one spectroscopic class hotter/bluer. They are 1.51 and 1.39 times the mass of the sun and orbit each other once every 2800 years.

The planet, Kepler 14b -- I am tempted at this point to make up a name for it. I dub it Keforb -- is 8.4 times the mass of Jupiter. This makes it a fairly large gas giant and definitely not habitable by human standards. Could one of it's moons be habitable? Well, let's keep looking at the planetary parameters.

The orbital period, the length of one Keforbian year, is 6.79 Earth days. That's pretty quick. If you've been playing along at home, you might remember that planets orbit faster the closer in to their star they are. Keforb is only about 0.08 AU from it's primary sun. (Mercury is about 0.4 AU from the sun.) I don't need to do any calculations to know that this planet is going to be way too hot for life, even if it had a suitably-sized moon. It's the sort of planet that's known as a hot Jupiter.

I found this pretty awesome NASA page which shows everything you need to know about Keforb (except for my awesome name) including, if you click on the buttons down the bottom, where Kepler 14's habitable zone is and the orbits of solar system planets for comparison.

So the habitable zone for this system is out past Mars's orbit, in the range of 2.17–3.56 AU, roughly where the asteroid belt is in our solar system.

Similar configurations?

OK, so there's no hope for life on the one known planet in the Kepler 14 system, but what about other possible planetary systems of similar configuration. I mentioned Tatooine last week as being similar to the planet in the Kepler 16 system. The Kepler 14 system is different because the planet orbits only one star, not both of them.

A system of this sort that pops up in fiction every now and again is a planet orbiting a red dwarf star which is in turn orbiting a larger, brighter star. I'm pretty sure I've read about this sort of thing more than once but the series that springs to mind first is the Second Sons Trilogy by Jennifer Fallon. Partly this is because it's one of my favourite series, so I may be biased. In the first book, Lion of Senet, we are introduced to a world with two suns: a small red sun and a more distant yellow sun. The world orbits the red sun and the red sun orbits the second yellow sun. The residents experience two different types of sun sets and sun rises and their "nights" are when only the red sun are up in the sky. They also get to experience side-effects from the tidal forces of a) having two suns and b) being in a relatively close orbit with the red dwarf. This means lots of volcanoes and tidal waves (I think the first book even opens just after a volcano, but I may be wrong and I don't have it on hand to check).

It's billed as a fantasy series, and it's certainly written in a fantasy style with feudalism and political intrigue and war and things (the rest of Jennifer Fallon's books are indisputably fantasy). However, there isn't any more magic than in our world which makes me tempted to say that it's technically sort of science fiction. But without laser guns (or any sort of guns. There are swords, though). Maybe "sword and science" rather than "sword and sorcery".

Anyway, I highly recommend it, not just because of the interesting celestial mechanics, but also because it's a damn good read. The hero, Dirk, goes around applying his brain, not brawn. Also, if you go in solely for the celestial mechanics, you might be disappointed because, while good for the reasons I've mentioned above, it's not quite perfect, mostly for reasons of plot.

Sometimes, it's best not to let science stand in the way of a good story. (So long as you try and at least some of the science is sensible.)

Weird Worlds: Kepler 16

I have talked about the Kepler exoplanet-finding mission in the past (planet spotting, Kepler 11, KOI 730). Today I'm going to look at a planet Kepler has found in a binary star system: Kepler 16(AB)b and next week I'll talk about Kepler 14b which is also in a binary system but in a different configuration.

A binary star system is where two stars are close enough together to be in orbit around each other, instead of individually orbiting the centre of the galaxy. Instead they orbit the centre of the galaxy together, much the same way as the Earth and the moon orbit the sun together. Also sort of like on Tatooine from Star Wars.

Kepler 16b

Artist's impression of the Kepler 16 system.
Credit: NASA/JPL-Caltech/R. Hurt

Basic stats for the Kepler 16 system are somewhat unartistically presented here. I'll go over some of them below.

As far as we know, there are three bodies in the Kepler 16 system: the two stars which are designated Kepler 16A and B (and together Kepler 16(AB). As per usual convention, the more massive star will be A while the smaller will be B. The usual convention, for non-binary systems, is that the star is designated Whatever-a and planets in order of discovery then mass (so mass when several are discovered simultaneously) are designated Whatever-b, Whatever-c etc. The lower-case a for the stars is a bit redundant here, though. Masses and distances for the system are as follows:

• Star A:
• Mass: 0.69 solar masses
• Size: 0.65 solar radii
• Temperature: 4450 Kelvin (about 4180º C or 7600º F)
• Star B:
• Mass: 0.20 solar masses
• Size: 0.23 solar radii
• Stars AB together:
• Orbital period: 41 days
• Orbital separation: 0.224 AU (for comparison, Mercury's orbit is 0.387 AU from the sun)
• Planet Kepler 16(AB)b:
• Mass: 0.33 Jovian masses
• Size: 0.75 Jovian radii
• Orbital period: 229 days
• Orbital radius: 0.70 AU (Venus is 0.72 AU from the sun)

It's not possible to separate out and measure the temperature of star B because it is completely overwhelmed by the light from star A. We can only make guesses based on it's spectral type which is determined from its chemical make up. Also note that the distance measurement for the planet is from the centre of mass of the two stars -- the barycentre.

For a bit of fun, let's work out how big and bright each star would be from the planet, relative to the sun. I should note that this is a gas giant planet and so any possible light would be more likely to exist on one of its moons, not the planet. The suns would appear the same size from a moon, though.

First, how big would the suns appear? (See this post for details on the calculation.) Remember for comparison that the sun (and moon) have an angular diameter of about 0.5º. So, sizes:

• Star A as seen from planet on average: 0.49º, so about the same as the sun
•  Range: 0.43º to 0.59º
• Star B as seen from planet on average: 0.18º, which is about a third of the diameter of the sun
• Range: 0.15º to 0.21º

Now, how much light would reach the planet from the two suns? Or how brightly would the stars illuminate the planet? This is a slightly more complicated calculation which you can read about here (and for comparative purposes, about 1400 Joules of energy hit each square metre of the Earth from the sun each second). Remember, also, that sometimes star B will pass behind star A, sometimes they will appear next to each other and sometimes B will be in front of A, blocking out some of A's light but contributing its own. I'm just going to calculate them separately.

• Star A: 410 Joules per square metre per second, which is a bit less than a third of the light energy the Earth gets from the sun. It is also going to be redder light, thanks to the cooler temperature of the star.
• Star B: assuming its temperature is about 2500 K which is within the plausible range for M type stars (star A is K type, if you're curious), I get about 5 Joules. That's 3-4% of the light from the sun hitting Earth. On the other hand, it's still a lot brighter than the full moon in Earth's sky (and significantly redder). About 1400 times brighter, in fact. And about as bright as 260 100 Watt light bulbs at a distance of 10 m. Except it would probably feel dimmer thanks to the redness.

Finally, let's suppose that there is an Earth-like moon of this gas giant planet Kepler 16b. How warm would the suns make it? The calculations are discussed in this post on the habitable zone. Since most of the warming comes from star A, I'm going to ignore star B for this calculation. The average temperature I get is about 190 K which is about -85º C or -121ºF. So a little bit too cold for life, but not too hot, which is a harder problem to solve. Maybe throw in a bit of greenhouse warming and a lot of snow gear and you have yourself a snowball planet with two suns. Not just cold, but pretty cool.

And if you want to read more, here is a nice New Scientist article about it. And, below, a video from APOD:

Let's talk seasons

Depending on where you live, you probably experience two or four seasons a year. Does this have to be the case on planets other than Earth? What causes Earth's seasons and what else might cause seasons on other planets? Read on!

Tilting

Earth spins about an axis that runs approximately from the North Pole to the South Pole. It completes one rotation per day (and, indeed, a day is defined by the period of rotation). the degree of tilt of the axis never changes. This is because one of the fundamental laws of physics is that angular momentum must always be conserved. If the angle of tilt changed then the direction of the angular momentum would also change and this isn't possible without some sort of external influence (like an asteroid, which isn't quite something we want to happen).

What is this angle of tilt relative to? Well, it's the amount by which the axis of rotation differs from making a 90º angle with the plane of Earth's orbit around the sun. Hopefully the image below helps.

Nabbed from Wiki. Credit: Dennis Nilsson, NASA.

As I said, the tilt doesn't change, so for part of the year the northern hemisphere is more exposed to the sun and for another part of the year, six months later, the southern hemisphere is more exposed to the sun. In temperate climates, these periods of greater exposure are called summer while the periods of least exposure are called winter. The in-betweens, as I'm sure you're aware, are spring and autumn. Here is another diagram to illustrate this:

Light and heat from the sun is hitting more of the southern hemisphere than the northern hemisphere.

Hot and/or tropical regions, which tend to lie close to the equator, experience two seasons: the wet season and the dry season. Similarly, the other extremes of the planet, the poles, also experience two seasons: polar day and polar night. This is because during winter the sun never rises the poles and never sets during summer. Within the polar circle but away from the actual rotational poles, there will be a period of transition between the two seasons (of varying length, depending on distance from the poles).

The greater the axial tilt, the more pronounced the differences between summer and winter will be (see the bit about Uranus at the very end for a very extreme case).

OK, so axial tilt causes Earth's seasons. What else can cause seasons?

Near and Far

Another possible cause of seasons is an elliptical orbit. This occurs when for part of the year a planet is noticeably closer to its sun than for the rest of the year. So when the planet is physically closer to the sun, the whole planet experiences summer (not just one hemisphere). During the more distant part of its orbit, the whole planet would experience winter.

On Earth, key seasonal dates, such as the solstices and equinoxes, are defined by the length of daylight (shortest day/night of the year and equal day and night respectively). On a planet with an elliptic orbit the key dates would be defined by significant points in the orbit. The equivalent of the summer solstice would be the periastron, the point of the planet's closest approach to the star. The winter solstice would be replaced with the apoastron, the time when the planet is furthest from the sun.

An interesting thing to note is that, thanks to Kepler's second law, a planet will move more quickly in its orbit when it's closer to its star than when it's further away. (If you follow that wiki link, there's a nice little animation which sort of explains it.) The result is that summer on such a plane will be briefer than winter. The more eccentric (non-circular) the orbit, the greater the difference between the lengths of summer and winter (and the closer the planet will be to its sun during summer). Could make for an interesting cultural interpretation of the seasons.

Pulsing star?

Some stars vary their brightness. Cepheid variables, for example, pulse with a regular period (which can be anywhere from one day to a few months). Theoretically, this could induce a seasonal variation for any surrounding planets. However, there is a problem when it comes to life evolving on such a planet. These stars are unstable and won't last very long (on an astronomical scale) in their pulsing state. This makes it a bit more difficult to justify having an inhabited planet around them. Maybe a planet with a colony that's studying the star. Anything more natural probably wouldn't last or might not have enough time to have evolved (depending on the size of the star). Of course, there's no rule saying impending doom couldn't be central to a plot.

Wacky planets

Uranus is sideways. (Also, the rings aren't
really red, just coloured that way for emphasis
here.) Credit: Lawrence Sromovsky, (Univ.
Wisconsin-Madison), Keck Observatory.
via APOD

Uranus is an interesting case. It's axis of rotation lies almost in the plane of it's orbit. So for part of the year the south pole points directly towards the sun and part of the year the south pole points directly away from the sun. In between is a transition similar to the type of seasons Earth experiences, but with more extreme beginning and end.

Of course, Uranus is too large and with too dense an atmosphere to support life as we know it. However, it's possible that a rocky planet more suitable for human habitation could also have this kind of extreme tilt. Everywhere except on the equator there would be periods of multiple days of darkness. Even on the equator, polar summer and winter would be spent in perpetual twilight.

I suspect this sort of configuration could also cause interesting weather/climate issues, but I think that would depend a lot on the atmosphere as well. Could be problematic.

Some Links!

Some interesting sciencey links, presented in the order in which I came across them (which I think is the order they were published anyway).

First up! This New Scientist article is about how the wakes of ships (the white "trail" they leave behind as they plough through the ocean) temporarily increases the albedo of the Earth—how much light from the sun it reflects. This would be good news for climate change (that is, enacting a more positive change and less the "oh gods we're all going to die in a tropical hell" sort) if not for all the greenhouse gases those ships also produce...

Next! This arXiv paper, "Kepler Exoplanet Candidate Host Stars are Preferentially Metal Rich", seems to back up some of the assumptions made in the paper I discussed a short while ago about the galactic habitable zone. In that paper they assumed that sufficient metallicity would be required to form planets and it looks like Kepler has confirmed that.

Penultimately! Forget about Martian meteorites containing fossilised life, simulations have shown that life from Earth may have made it off world and could be set to land on another planet. Could it be that when we finally get around to drilling through the Europan ice, we find Earthly extremophiles instead of aliens? Cosmos magazine have the full story.

Finally! National Geographic have an article about a newly discovered planet which has the potential to be Earth-like. Seems that it gets less energy from its sun than Venus does from Sol, but more than Earth receives. It's habitability thus depends on the appropriate weather conditions but with surface gravity 1.4 times Earth's it should be more or less possible to walk on the surface.

Weird Worlds: Gleise 581

A quick note before I jump into the meat of this week's post. From next week, I'm going to be on holiday for about two weeks. I'm going to try to queue up a couple of blog posts to self-update while I'm gone, but if something goes wrong I may not be able to do anything about it until I'm back in normal-internet land.

So this week, I thought I'd do another entry in my Weird Worlds series. This time about a planetary system which has gained quite a bit of media attention: Gliese 581. (Pronounced something like "glee-zeh", if you're curious.)

You may have heard of Gliese 581 discussed in the media with phrases like "second Earth" or "super Earth in the habitable zone" being thrown around. So how like Earth are we talking? Which part of the habitable zone? What sort of star is Gliese 581 anyway? Read on!

The Star

First the basics: Gliese 581 (which I will now refer to as Gliefeo because I am lazy and typing numbers is annoying it rolls off the tongue better) has four confirmed planets and two unconfirmed planets. For the sake of not getting too carried away I will focus on the four confirmed planets, although it was one of the unconfirmed ones which drew a large slice of media attention.

Gliefeo itself is a red dwarf star only about a third of the mass of our sun and a bit less than a third of the size. This means that to be within the habitable zone, its planets need to be significantly close to it than Earth is to Sol. Furthermore, the stellar environment these planets will be living with is very different to ours. Here is a nice Space.com article about it.

All of Gliefeo's planets were detected using the radial velocity method.


The Planets

As I mentioned above, Gliefeo's four confirmed planets are all quite close to their sun. In fact, as the image below (taken from my favourite exoplanet app) shows, all four are well within the orbit of Mercury (the grey circle). And the second image below (also taken from my favourite exoplanet app) shows two blue circles indicating radial distances of 0.1 AU and 0.3 AU (an AU is the distance between Earth and sun).

Gliese 581 and its four confirmed planets designated, from innermost to outer: e, b, c, d. The grey outline shows where the orbit of Mercury would be if Mercury's orbit were picked up and plonked around Gliese 581. The green annulus shows the habitable zone for Gliese 581 and both Gliese c and d are within the habitable zone for at least part of their orbits. Image from Exoplanet iOS app by Hanno Rein.

A representation of the Gliese 581 system. The inner blue outline denotes an orbit of radius 0.1 AU and the fainter outer circle (click to enlarge) shows an orbit of radius 0.3 AU. The four planets from inner to outer are designated e, b, c and d. Image from Exoplanet iOS app by Hanno Rein.

Some technical details on each of the planets (all taken from the Extrasolar Planets Encyclopaedia) :

• Gliese 581 e:
• Furthest distance from Gliefeo: 0.03 AU
• Length of year: 3.15 Earth days
• Mass: 1.94 Earth masses
• Gliese 581 b:
• Furthest distance from Gliefeo: 0.041 AU
• Length of year: 5.37 Earth days
• Mass: 15.64 Earth masses = 0.91 Neptune masses = 1.08 Uranus masses
•  Gliese 581 c:
• Furthest distance from Gliefeo: 0.07 AU
• Length of year: 12.9 Earth days
• Mass: 5.36 Earth masses
• Gliese 581 d:
• Furthest distance from Gliefeo: 0.22 AU
• Length of year: 66.8 Earth days
• Mass: 7.09 Earth masses = 0.41 Neptune masses = 0.49 Uranus masses

The last two planets, c and d, are the most interesting because of their location in the habitable zone.

Habitability

So are the two outermost planets of Gliefeo habitable? Well, probably not. In fact, chances are, none of these planets are able to support Earthlike life. Ignoring the issues with the star itself, as mentioned in the Space.com article I linked above, there are problems with all four planets.

Gliese 581 e is the closest to Earth in mass which, depending on its size (and composition) could mean that we could safely walk upon its surface from a gravitational point of view (well, safely is relative, but we probably wouldn't die instantly). However, it's so incredibly close to the star that it would definitely be to hot to support life.

Gliese 581 b is the largest of the four at around the size of Neptune. This almost certainly makes it a gas giant with a very dense atmosphere that would crush us if we tried to find a surface. It would also be very hot, not just because of it's proximity to the star, but because of the insulating effect of that thick atmosphere.

Gliese 581 c skirts the inner edge of the habitable zone, probably making it too warm for comfortable life. It's heavier mass also suggests a thicker atmosphere than Earth's (although this is purely speculation) and it could be more similar to Venus in terms of climate. That is to say: very inhospitable. It's surface gravity would probably also be a bit too strong for us, though this would depend a bit on composition.

Gliese 581 d falls into the class of planets somewhere between Earthlike and (mini) gas giant. We're not completely sure at what mass point planets stop being rocky and turn into mini gas giants.

(Side note: the mini and the giant should really cancel out, shouldn't they? Maybe we should call them gas balls to distinguish from larger gassy planets like the gas giants Jupiter and Saturn. But then, where would you draw the line? I suspect it would end up depending on the composition of the gas at lesat partially.)

So while Gliese 581 d spends roughly half its time in the habitable zone and half beyond it, it's probably not inhabitable itself. However, if it had any rocky moons like the solar system gas giants we know and love, those moons have a reasonable chance of being habitable. With the added advantage that they're going to be (probably) tidally locked to the planet, not to the sun, allowing the sun to more evenly warm its surface.

It's interesting, really, that of all the exoplanets discovered so far—not just around Gliefeo—the area most likely to be habitable is a moon of a gas giant. Really that says more about our detection techniques than anything else, but it does suggest some interesting possible world building.

Requests?

As I mentioned in the intro, I'm going away and hope to have some blog posts prepared in advance. However, my brain is all used up organising travel-related things without much room left for thinking about stories or reading new articles (my primary sources of blog post inspiration). So if any of you lovely readers have any requests for future posts, drop me a line in the comments!

Weird Worlds: KOI 730

Before I get onto the main part of the post, I'd like to apologise for the lack of short posts over the weekend but life's been hectic. Part of the reason for this is that I'm going to be away at a conference next week. I plan to queue up a post to automatically go live next Wednesday since I'm not sure how reliable my internet access will be (also because I'm not taking my laptop and will be relying on Blogger's willingness to talk to my iPad, another thing of which I am not confident). On the other hand, said conference should give me lots of fodder for short, if not long, posts. So that's something to look forward to.

On to the topic of the week! Today I am going to be writing about another crazy exoplanetary system. However, unlike Kepler 11, this one isn't quite confirmed yet. KOI stands for Kepler Object of Interest and means that it's a system that the Kepler mission has identified as potentially containing some planets (four in this case) but they haven't been confirmed by other supporting data. Because science is all about the independent evidence. Nevertheless, this is a blog about science fiction, so we are quite at home to a little speculation. As such, please remember that all of the facts I state below about planets are as yet unconfirmed and haven't quite passed into official scientific cannon.

Kepler Object of Interest 730

First, some basic facts about the system. The star has designations KOI 730 or KIC #10227020 (one of these is easier to remember than the other, so guess which I'll be using). It is similar to the sun, but slightly larger and slightly cooler (by a couple of hundred degrees). It is more than 4200 light years away. The four planets are all very close in to the star, much closer than even Mercury's orbit in the solar system, making them conclusively uninhabitable to life as we know it.

Here is a screenshot illustrating the system from this Kepler Candidates Exoplanet app (not to be confused with the other one I've referenced before which was of confirmed planets, albeit otherwise pretty much identical).

The KOI 730 system side on (as would be seen by the Kepler telescope itself). Three of the four planets are visible in this illustration. The white line indicates the plane of the orbit (actually, it's a bit of a trail/track following each planet, but that doesn't come across too well side on). The planets are to scale relative to each other but not relative to their sun. Also, I don't think they get smaller when their on the opposite side of the star to the observer.

Resonating

That's OK, because that's not the particularly interesting thing about this system. The scientifically interesting thing is that this system is locked in an orbital resonance. An orbital resonance is when two planets (or moons) orbit in such a way that they both complete an integer number of orbits in the same length of time. (An integer is a whole number such as 1, 2, 3, etc.) Some examples from the solar system are the Jovian moons Io, Europa and Ganymede, which are locked in a 1:2:4 resonance of orbital periods. This means that Ganymede's and Europa's periods (how long it takes them to complete one orbit around Jupiter) are respectively four times and twice the period of Io. (Sometimes this might be written 4:2:1 indicating that Io makes completes four orbits in the time it takes Europa to complete two and Ganymede to complete one. It depends on the convention being used.) Another example is Neptune and Pluto locked in a 2:3 period orbital resonance (so Neptune completes three orbits in the time it takes Pluto to complete two). A different sort of resonance is that experienced by Mercury, which completes two orbits in the time it takes to make three revolutions (so three Mercurian days are equal to two Mercurian years).

Orbital resonances can either make the system, or more specifically, the bodies involved in the resonance, stable or unstable. Yep, I know that sounds like they don't do anything because both possible outcomes are covered, but that's not true. What I mean is that if the resonance is unstable, you get things like gaps in the rings of Saturn (caused by some of Saturn's moons). On the other hand, the Jovian moons I mentioned before are in a stable resonance and Mercury is in its spin-orbit resonance rather than being tidally locked thanks to the gravitational tugs of other planets.

Which brings me to some of the effects of orbital resonance. Bodies locked in an orbital resonance exert a greater gravitational influence on each other than they otherwise would. For example, when Ganymede, Io and Europa are all lined up (Io and Ganymede on one side, Europa on the opposite side of Jupiter), they all experience a heightened tidal effect as the gravitational pulls of the other two planets add directly to the pull of Jupiter, causing additional friction in the planet interiors (and, for example, contributing to Io's volcanism).

Back to KOI 730

So I mentioned that KOI 730 has four planets and that these are locked in an orbital resonance. According to the first articles I read about it (in New Scientist and somewhere else I can't recall) and the original paper (section 5.3 is specifically about KOI 730) the resonance scheme for KOI 730 is 6:4:4:3. Notice the two fours there? That is why there were a spate of pop science articles about this system. Those two fours indicate that two of the planets are in the same orbit since orbital periods depend only on the star's mass and the distance of the planet from the star.

Two planets in the same orbit. They're located at two of the Lagrange points you might remember me mentioning a while back. Due to their positioning, they are known as trojan planets after the trojan asteroids that follow and precede Jupiter and Saturn in these same Lagrange points. The two planets are 118º apart along their orbit and are slowly, over millions of years, edging towards each other (at least; the authors of the paper speculate that they might last billions of years).

But yes, eventually they will collide.

The configuration of these two planets, KOI 730.02 and KOI 730.03, is such that they form an equilateral triangle with the star as the third point. Unfortunately, this puts the second planet as far away from the first as the sun, making it about a quarter of the height of the full moon as seen in Earth's sky. It would also always be about two-thirds illuminated and it wouldn't move around in the sky relative to the stars. If the planets were tidally locked to their sun then the other planet would stay locked in the same position in the sky while the stars moved around it, which would be fairly cool to observer. (Just think of the mythology that could arise surrounding that set up!)

It also bears mentioning that one of the theories of Earth's creation has two proto-planets forming at Lagrange points like these trojan planets. The other proto-planet, usually labelled Theia, and proto-Earth, inched towards each other and eventually collided, merging and splashing, so to speak, to form Earth and moon as we now know them.

Later, when I googled this again in preparation for writing this blog post, I found this article from Sky & Telescope, which features one of the authors of the original paper saying that further analysis of the data suggests a 8:6:4:3 resonance might be more fitting. This turns one of the trojan planets into a different orbit, farther out, and makes the system less exciting. I mean, a system in which all the planets are in resonance with each other is still pretty notable, but it's just not quite as imagination-grabbing as TWO PLANETS IN THE SAME ORBIT. Although, there's still potential for interesting story-science there when the planets line up and whatnot.

Oh well, co-orbiting planets in KOI 730 or not, the concept was around before this paper was written and there's no firm reason to not suppose we couldn't have trojan planets somewhere else. Maybe a gas giant with, instead of trojan asteroids following/leading it around, a full-sized terrestrial planet. Or two terrestrial planets sharing a habitable orbit...

The possibilities are endless. And science is cool even when it's speculative.

Weird Worlds: Kepler 11

This week I decided to talk about some of the weird, wacky and real exoplanets that we have discovered. I am going to just focus on a few with possibly more to come at a later date.

Kepler 11

I have mentioned it in the past and the Kepler Mission is certainly the current poster-child of the planet-spotting community. It has been staring at a patch of sky near celestial north since 2009 and in that time has found 1235 potential planets, 16 of which have been confirmed and most of which require further investigation.

There are six confirmed planets in the Kepler 11 system (the star, which was too far away and hence dim to have a better name, is designated Kepler 11). The star is similar to our sun. The planets orbiting it, however, are nothing like our solar system. Five of the planets are within what would be the orbit of Mercury if they were in our solar system. The outermost (or at least, the outermost discovered so far) is just outside the orbit of Mercury. They are also all relatively small, on the scale of previously discovered exoplanets, but much larger than our rocky worlds. They range in mass from 2.3 times the mass of Earth to slightly less than the mass of Jupiter (about 300 times the mass of Earth) and size from 1.93 Earth radii to 0.4 Jupiter radii or about 4.4 Earth radii (interestingly, the smallest isn't the lightest—Kepler 11-f is much less dense than Kepler 11-b). Their years range from 10.3 to 118.4 Earth days. For comparison, Mercury's year is about 88 Earth days.

Let's ignore the fact that all of these planets are too close to their sun to be habitable and have a think about what such a system would look like from within. From Earth, all the planets out to Uranus just look like bright stars which move across the sky. How different would the night sky look from a planet in the Kepler 11 system? For no particularly compelling reason, I'm going to work this out for Kepler 11-f which is the least massive and the second furthest out from the star.

First a picture. It's a (cropped) screenshot from this Exoplanet iPhone app, showing the Kepler 11 system. The planets aren't to scale with the star (they are very much inflated), but are to scale with each other.

Obviously the trails following the planets aren't real either, but are there just to give you an idea of the orbits. I like how the developer made it so only the side of the planet facing the star is lit up. Yay, realism.

Now some stats on the system, which I have mostly gotten from that Exoplanet iPhone app. Also, I'm going to abbreviate the names to Keb, Kec, Ked, Kee, Kef, Keg for Kepler 11-b through f. Because I can. (And if you're wondering, there's no Kepler 11-a because that designator is reserved for the star.)

• Keb: 
• 4.3 Earth masses, 
• 1.93 Earth radii, 
• orbit is 0.091 AU, 
• closest approach to Kef is 0.159 AU
• Kec: 
• 0.04 Jupiter masses = 12.7 Earth masses, 
• 3.09 Earth radii, 
• orbit is 0.106 AU, 
• closest approach to Kef is 0.144 AU
• Ked: 
• 6.10 Earth masses, 
• 0.31 Jupiter radii = 3.41 Earth radii, 
• orbit is 0.159 AU, 
• closest approach to Kef is 0.091 AU
• Kee: 
• 8.4 Earth masses, 
• 0.40 Jupiter radii = 4.4 Earth radii, 
• orbit is 0.194 AU, 
• closest approach to Kef is 0.056 AU
• Kef: 
• 2.3 Earth masses, 
• 2.56 Earth radii, 
• orbit is 0.25 AU
• Keg: 
• 0.95 Jupiter masses = 301 Earth masses, 
• 0.33 Jupiter radii = 3.6 Earth radii, 
• orbit is 0.462 AU, 
• closest approach to Kef is 0.212 AU

Now for some calculations. Using the formula from my very first blog post, I calculated how big the other planets would appear at their closest approach to Kef. Also how big the sun, Kepler 11 (or Kea, why not) would appear. For all of these, I'm going to list them in units of the diameter of the full moon, except for Kea which I list in units of the diameter of the sun (because it's a star. Actually the angular diameter of the sun and the moon are roughly equal, which is why we get such nice looking solar eclipses). Also, this wiki table is useful for comparing how big things are in Kef's sky with how big things are in our sky. For future comparison, Venus varies in size from 0.16'-1.1' where the ' indicates arcminutes and there are 60 arcminutes in a degree. At it's closest visible point, however, Venus is a crescent that size, and full only at it's farthest point. That's about a thirtieth (0.036) of the size of the full moon. Easily resolvable with a telescope, but not quite resolvable as a disc with the naked eye.

At their furthest from Kef, all 5 other planets are similar to the size of Venus at it's furthest (and also it's fullest since that's when they are on the opposite side of Kea to Kef). At their closest approaches, however, there's a bit more variation. Remember for all of these except Keg the planets would be crescents of this size and only Keg would be full. Furthermore, I do declare that 1 FM (full moon) = 0.5º, and is the units I've used below. Because I didn't think anything would be gained by copy-pasting  "times the size of the full moon as seen from Earth" five times. (This is totally how units are invented. ;-p ) I may have also included some of the values as fractions.

• Keb: 0.05 – 0.1 FM (max is a tenth)
• Kec: 0.08 – 0.2 FM (max is a fifth)
• Ked: 0.08 – 0.3 FM (max is approximately a third)
• Kee: 0.1 – 0.7 FM (max is around two thirds)
• Keg: 0.05 – 0.15 FM (max is almost a seventh)

I'm not really sure about Keb, Kec and Keg—I suspect they may just look like very bright stars—but the other two would definitely be visible as circles/crescents to the naked eye. The night sky of Kef would be a very different place to ours. I can't help but think that with such obvious neighbouring planets any intelligent life that evolved on Kef would work out celestial mechanics rather more quickly than we did. After all, we had to really pay attention to the sky to notice that some of the shiny points of light moved while others didn't. Not that life as we know it is likely to have evolved on Kef. Which brings me to the day sky.

Kef's sun, Kea, would be 4.7 times the size of Sol in our sky. Almost five times as big. (For those keeping track, that's the same as saying almost five times the size (width) of the full moon. That might actually be easier to visualise since it doesn't generally hurt to look at the moon.) It would also be around 22 times brighter than our sun, giving Kef 22 times as much energy as the sun gives Earth. Do you see why Kef might not be particularly habitable? Kind of like how you wouldn't want to live on Mercury, but more so.

Also, as you may have surmised from the first set of data I dumped, Kef isn't very dense. Density is given by mass divided by volume and volume is proportional to the radius cubed (actually, for a sphere, volume is ⁴/₃ π R³). So Kef is only 0.14 times the density of Earth and, if it had a solid surface, which is highly unlikely, it would have a surface gravity of around a third of Earth's (similar to Mars's), despite being more massive. It's actually about three quarters the density of water, which isn't that surprising because if it were made of water (and I have absolutely no idea what it's composition is), we wouldn't expect water to be liquid in those conditions. What's interesting is that of the other planets, the two innermost are comparable to Mars in density (less dense, but close), the next three, Ked-f are comparable to Saturn (Saturn actually lies between Kef and Kee with Ked being a bit denser) and the outermost, Keg, is ridiculously dense. Like, denser than Osmium, the densest natural element. I think what this really highlights is how we only have an idea of planet size from the amount of dimming we see when it passes in front of the sun. The mass of the planets is worked out from the gravitational tugs they exert on each other in multi-planet systems (for transiting planets like the ones Kepler is looking for, that is). I'm willing to believe the densities of the other 5 planets, but I think I'll wait for better data before believing Keg.

My original plan was to talk about a variety of weird worlds in this post, but I got a bit carried away with Kepler 11, it seems. Stay tuned for more weird worlds in the future.

Moon-spotting possibilities

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.

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
• 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.

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.

Introduction to Gravity

I thought I'd do a series of posts on gravity because it's a huge topic and kind of important. I don't really want to do all of these in a row so if any of you have requests or suggestions for topics, please let me know in the comments. :-)



Newton's law of gravity



Almost everyone has heard the story of Newton sitting under an apple tree and being inspired towards understanding gravity thanks to a falling apple. You may have heard the version of that story where the apple falls on Newton's head but I recently read that it actually landed next to him. I am more inclined to believe this version of the story because I know that if an apple fell on my head I'd be too busy cursing the tree to have a flash of inspiration.

What was the revelation Newton had about gravity? Well basically, he gave us a simple, universal description of gravity. The same force that causes the apple to fall towards the ground also keeps the moon orbiting the Earth and the Earth orbiting the sun. The mathematical expression for the force of gravity between two objects that Newton left us with is:

F is the force of gravity between two objects of masses M and m kilograms, with their centres separated by distance r meters. G is the gravitational constant and is equal to 6.67 × 10^-11 km³ kg^-1 s^-2 .

However, in our day-to-day experiences, it is not forces, per se, that we are most aware of but accelerations. For example, if you are on a train moving at a constant speed (not accelerating, that is; not slowing down or speeding up), you can't tell how fast you are going solely from its motion. The only thing that would give it away is the bumping up and down thanks to uneven tracks. However, it's easy to tell when the train is slowing down or speeding up because you are either pushed backwards or forwards in your seat (depending on which way you are facing).

Although gravity is always pulling us towards the centre of the Earth, what we actually feel is the ground (or floor or chair or whatever) holding us up and preventing us from falling towards the centre of the Earth. It's gravity accelerating us into the floor that we experience as weight. If there was no floor and we were just falling, we would actually feel weightless (air resistance notwithstanding). The International Space Station is well within Earth's gravitational field but it is constantly falling, which is what makes the astronauts inside feel weightless. Luckily it's not just falling straight down; it also has a horizontal velocity which means that in the time it falls downwards a certain amount, the curvature of the Earth results in the ground also falling away by the same amount. This is called an orbit, and I will talk more about them in the next section.

Back to the equation above. How do we turn this into what we experience every day on Earth and then into what we would experience if we were living on another planet. The important quantity to calculate is g, the acceleration due to gravity. Every force (or sum of forces) can be described as an acceleration acting on a mass. F = ma is the famous equation and is the mathematical representation of Newton's second law (side fact: Newton's law of gravitation didn't get a number, his three laws refer to more basic laws of mechanics). To find acceleration due to gravity on the surface of a planet, we equate F = mg (rather than F = ma, since g is the symbol we use for acceleration due to gravity) with Newton's law of gravity:

Acceleration due to gravity, g, depends only on the mass of the body exerting a gravitational force and on the distance from the centre* of that body. All bodies, no matter what their mass, m, will accelerate at the same rate in the same gravitational field.

To find g at the surface of the Earth, you need to substitute in the mass of the Earth for M and the radius of the Earth for r. On Earth, g = 9.8 meters per second per second, which means that, if we ignore air resistance, something that is falling will gain 9.8 m/s of speed each second. It also means that the force with which we are constantly pushed into the ground/chair/bed is equal to our mass times 9.8.

On planets other than Earth which are smaller, larger, heavier or lighter, there are different accelerations due to gravity which we can find by throwing the right numbers into the equation above. Some examples from rocky† planets and moons in our solar system (where g is acceleration due to gravity on Earth's surface):

• Moon: 1.7 m/s² = 0.17 g (about a sixth of Earth's gravity, so you would feel a sixth as heavy.)
• Mars: 3.7 m/s² = 0.38 g (between a third and two fifths of Earth's gravity)
• Mercury: 3.8 m/s² = 0.39 g (coincidentally very close to Mars)
• Ganymede: 1.5 m/s² = 0.15 g (also about a sixth of Earth's gravity)

And because this is easily applicable to extrasolar planets — that is, planets outside of our solar system — I have also calculated some surface accelerations due to gravity for a few known exoplanets (links below are to Wiki, but I got my values from the Exoplanet iOS app; see below).

• Gliese 1214 b: 8.6 m/s² = 0.88 g (Just under nine tenths that of Earth. However, it's just inside its host star's habitable zone (the star is called Gliese 1214) which means it'll probably be too hot for human habitation. It could even have a runaway greenhouse effect like Venus. It's also possible that this planet has a very thick atmosphere making it more similar to a small gas giant like Neptune than to Earth.)
• CoRoT 7 b: 18.4 m/s² = 1.9 g (Just under twice Earth gravity so you would feel almost twice as heavy and, more vitally, your organs would all press down on each other twice as strongly. According to NASA (pdf, sorry), this isn't terribly sustainable in the long term for humans as we are now. Personally, I don't think it would take an awful lot of genetic engineering to fix this for us (we are talking science fiction, after all). The bigger problem with this planet is that it's much to close to its sun for our comfort or survival.)
• Kepler 11f: 3.4 m/s² = 0.35 g (About a third of Earth's gravity. Compare with Mars or Mercury. Unfortunately, it's also slightly too close to its star to be habitable. Incidentally, the whole Kepler 11 system is quite interesting with six confirmed planets so far.)

(If you have a hankering to include some real exoplanets in your story, I highly recommend this iOS app is an excellent resource. It is a frequently updated database of all the confirmed exoplanets that have been discovered, including their statistics (mass, distance from star, radius where available, whether it's in its star's habitable zone...) and you can even pan through and around a zoomable 3D map of the Milky Way. And when I say zoomable, I mean you can zoom right in to see the planets orbiting their stars at the correct distances and with the appropriate relative velocities. Even if you don't care about the specifics of the planets, that Milky Way map is worth the time it takes to click the free download link. For the record, I am in no way connected to this app, I just think it's awesome.)


* Technically, the distance from the centre of mass, but for round or roundish things like planets the centre of mass is generally the centre of the planet.
† Rocky because you can't stand on the surface of the gas giants. It's possible to calculate the acceleration due to gravity experienced by a hovering platform or similar, however.

Orbits

As we've established, gravity is what keeps things in orbit around other things. As such, we need to use what we know about gravity to work out how fast something has to orbit for different distances and masses of objects. In fact, Kepler had worked this out to some degree before Newton came along, but Kepler's third law was slightly less specific than could be calculated using Newton's law. Kepler realised that for orbits, the ratio between the cube of the semi-major axis and the square of the period was constant. The semi-major axis is the same as the distance to the larger body from the smaller (like r in the equations earlier) for circular orbits and half the length of the longest side of an ellipse (oval). the period, T, is the time taken to complete one orbit, so if we're talking about planets, then it's the length of a year. We can derive the constant part of Kepler's third law using Newton's law of gravity and the equation that describes centripetal force.

As it happens, I talked about a lot of the ingredients for working out how fast a planet should orbit around its star in my recent post about space elevators. The set of equations below starts by equating the centripetal force (the force required to keep something of mass m moving around in a circle with radius r and at speed v) with the gravitational force (centripetal on the left, gravitational on the right of the equals sign), then shows the derivation of Kepler's third law (the second last line) and finally gives the period, T, of a planet orbiting around a star of mass M at a distance r. Feel free to let your eyes glaze over if maths isn't your thing. You have been warned.

The final line gives us the period in seconds, which for most things isn't terribly helpful. To find out what the length of your planet's year is in days or Earth years you will need to divide your answer for the period by 86400 or 3.15 × 10⁷ respectively.

The reason I chose to rearrange Kepler's law to solve for the period rather than for the semi-major axis is because for science fictional purposes, the distance from the star is more likely to be fixed for plot purposes. For example, an human-inhabited planet has to be in the habitable zone. Exactly what the habitable zone is will be, I think, the subject of a future blog post.

I should also point out that these equations are completely applicable to man-made satellites or moons as well, you just need set the planet's mass to be M instead of the star's mass. Just remember that if your comms satellite is x km above the surface of the Earth/whatever planet, you have to add on the radius of the planet to find r to throw into the equations I've talked about today.