Year-long days and living in them

This blog post was inspired by an email conversation with someone regarding the possibility of a planet having year-long (or half-year long) day/night cycles. The original question was whether this is even possible and whether such a planet would be habitable.

From a purely astronomical point of view, this is definitely possible. There's no reason why you couldn't have a slowly rotating planet at around the same distance from it's sun as Earth is (well any reasons that do exist are fairly theoretical so we can ignore them). That said, if the planet is similar to Earth and its sun is similar to ours, then you kind of have to have the same length year because the length of the year (ie how long it takes to orbit the star) depends only on the mass of the star and the distance from it. This is due to Kepler's Laws, which I have previously discussed here. If you made no changes to star/planet distance, the year length would have to be the same.

Image nicked from Wiki here. The little red line
represents the same point on the surface of
Mercury. The numbers are the order in which
the positions happen: 6, 1, 2 are night for
the red line and 3, 4, 5 are day, roughly.

You could also have something similar to Mercury which has three rotations (called "sidereal days" which are measured relative to the stars, not the sun) to two years. Because it rotates so slowly, weird stuff happens with its solar days (the light/dark periods, completely ignoring the positions of stars) so that in one year it experiences half a solar day. Mercury is like this because it's so close to the sun. It could have been tidally locked (the same side always facing the sun – discussed further, including for Mercury in particular, here) but the gravitational effects of the other planets in the solar system caused this more unusual resonance.

However, if we're talking a planet as distant from the sun as Earth is, there's no danger of it becoming tidally locked in the sort of cosmological time frame we're currently living in. The time taken for the angular momentum between planet and star to be distributed into the tidally locked configuration takes longer the further apart they are (and the less massive when they're close enough). The Earth-moon system will eventually become more tidally locked: the moon already faces the same side towards us all the time, and eventually the same side of Earth will always point towards the moon.

But that's a bit of a tangent, back to planets with long days and nights. You could have a planet rotating as slowly/quickly as you like, but you should be mindful that the people living there would almost certainly have a way of distinguishing between sidereal and solar days. Ancient people on Earth already had this worked out (the difference between sidereal and solar days is why the stars move across the sky with the seasons).

Living there

Uranus: almost completely sideways.

If you did have a planet with a year-long day, the periods of day and night would be roughly equal in the same way they are on Earth, just scaled up. It could vary a bit depending on the planet's axial tilt (how much the line between the poles is tilted relative to the plane of it's orbit — Earth's is around 23º and changes slightly when earthquakes occur) so the more inclined the axis, the more extreme the seasons. If there was no or very little axial tilt, there wouldn't be seasons. The other variable in day/night lengths is the latitude. Further away from the equator sunrise and sunset would last longer and the shortness of winter days and length of summer days would be more extreme (as on Earth, but a different axial tilt could make this more so). If there was no axial tilt, the poles would be in a state of twilight permanently. The other extreme is something like Uranus which has a 90º-ish axial tilt so that during a southern summer the south pole points towards the sun and during a southern winter the south pole gets no sun at all. Spring and Autumn are the transition period. The equator is in twilight during summer and winter and has more "normal" days, like what we're used to, during spring and autumn.

Also, astronomical plausibility aside, I'm not convinced complicated life could naturally arise on a planet with a super-long day/night cycle, due to the long periods of boiling (day) and freezing (night). In terms of temperature-stability, probably only the twilight areas would be habitable. I suppose you could have migrating species (but that also has problems because in staying in permanent twilight they'd need sufficient landmasses connecting the two poles). Also, you'd probably get some sort of storms around the twilight zone, since the temperature would be in in a state of flux. I'm not an expert on atmospheres or meteorology, though, so that's a (-n educated) guess and I can't be too specific. But in short: our 24 hour days are what keeps Earth's temperature relatively temperate and suitable for life.

There's be fewer issues for microbial life to arise but I don't know that anything larger would be viable. Maybe at the poles: if the planet was slightly closer to its star than Earth is, there could be non-migratory life living near the poles and with a stable orbit and rotational period, it should survive. Since the non-polar regions wouldn't have naturally arising complex life, there could be with completely different ecosystems/forms of life at either pole with only something like microbial ancestors connecting them.

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.

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.

• 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.

Gravity: Relatively general space

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

Curvy

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

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

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

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

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

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

* miscellaneous proposed but untested theories notwithstanding.

Black as black

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

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

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

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

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

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

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

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

Barely scratching the surface

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

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

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.