Habitable solar systems

The galaxy
Most galaxies fall into three kinds: irregular galaxies are small and formless masses of stars, rich in dust and hydrogen and containing mostly younger stars (see below for star classes); conversely, elliptical galaxies don't have much hydrogen or dust, and their stars tend to be very old; finally, spiral galaxies (such as our Milky Way and our closest neighbour, the Andromeda Galaxy) have a complex structure similar to a pinwheel, with a mixture of both young and old stars. About half of the galaxies (but none of the spirals) are dwarf galaxies. (Notes: Ms is the mass of the Sun, about 2.0 × 1030 kg; Ls is the luminosity of Sun, about 3.8 × 1026 J)

Most likely, life needs heavy elements, a liquid medium and probably a solid ground (also see here and here): only hydrogen and helium, produced by the Big Bang won't be enough. Element heavier than helium, from lithium to iron, are produced by nuclear reaction in the stars, while the rest of the natural elements (until uranium) derive from supernovae.

Dwarf galaxies are very poor in heavy elements, and they can be ruled out, and so, partially, irregular galaxies. Most of the stars in elliptical galaxies, being very old, were born right after the Big Bang, and didn't benefit from an interstellar medium already rich in heavy elements as the younger star did; it seems, therefore, that spiral galaxies are the best chance for complex chemistry.

Once we have chosen a galaxy, we need to find the "habitable zone" in which to search a life-bearing solar system. A large number of red and yellow stars, mixed with dust and gases, form the Disk, comprehending in turn a number of arms highlighted by hotter blue and white stars. The Halo, a spherical region around the main disk, contains only old, cold red stars very poor in heavy elements useful to build a rocky planet (carbon, silicon, iron).

These are much more abundant nearer to the Core, a region extremely rich in young, hot stars. On the other hand, the core is also richer in ionizing radiations, which disrupt atomic bonds and prevents the synthesis of complex molecules; while they'd not be much stronger than the radiation that a planet receives from its own star, the core also presents a higher rate of supernovae within a distance of few light-years.

Life could appear in the outer rim of the Core, though not in its centre: planets there would constantly be enlightened, day and night, by the huge number of stars. Still, the most likely location is the Disk, both inside and outside thae arms. In the Milky Way, the best distance from the centre is thought to be 25000-31000 light years (8-10 kiloparsecs), where most stars are between 4 and 8 billion years old.

Here there is a high-detail map of the Milky Way, produced by National Geographic.

Stars
All stars can be classified into spectral classes on the basis of their temperature, which directly affects their colour. From the hottest and brightest to the coldest and dimmest, the most important classes are called O, B, A, F, G, K and M. They're in turn divided in subclasses, identified by numbers - a star with a number between 0 and 4 is called an "early" member of its class, while a star with a number between 5 and 9 is called a "late" member (for example, as a G2 star, the Sun is an early class-G).

A second number, in roman numerals, identifies the star's luminosity class: 0 for hypergiants, I for supergiants, II for bright giants, III for normal giants, IV for subgiants, V for main-sequence stars such as the Sun, VI for subdwarfs and VII for dwarf stars.

Most of the other characteristics can be inferred from the spectral class, as shown by the tables below: (Note: Ga = gigayear = a billion years; mass in Ms, radius in Rs, luminosity in Ls)

In about 91% of all stars, size is a function of temperature: the line they call on a diagram is called main sequence. In this line, hotter stars are also extremely large (blue giants), while colder stars are small and dime (red dwarves). Stars are born as contracting spheres of gas; after the brief T Tauri-stage, in which a good part of the matter is thrown off, and the luminosity varies quickly. Growing older, stars pass through the main sequence, burning hydrogen into helium, and, depending from the mass, they can undergo different changes: Given the violence of these events, the lack of energy of white and black dwarves and the instability of supergiants and T Tauri stars, it's safe to assume that only stars in the main sequence (and, perhaps, smaller giants) can be suitable for life as we know it. Also, we can exclude O, B and probably A and early F stars as abodes of life: while extremely rich in energy, they burn out in a few millions years, far too little for complex chemistry, life and especially intelligence to occur in their system. M-class stars are a dubious candidate: they last long, but a planet needs to be very close to them to receive a substantial amount of energy (see here a more extensive treatise).
 * When hydrogen is running low, the pressure from the radiation at the centre decreases, and the star begins to separate into two layers: a wide diffuse shell and a hot core where helium is burned into carbon, and carbon into oxygen. The star swells up, becoming a red giant.
 * A star with low mass, such as our Sun, slowly dissipates the shell of hydrogen until only the core remains, becoming a white dwarf. This slowly fades to a cold black dwarf - not a star anymore.
 * A star with higher mass (above 8 Ms at the beginning) continues to burn its elements in different layers, burning carbon and oxygen into neon and magnesium, these in silicon and sulfur, and these in iron. At this point, it becomes a supernova, which explodes, producing all heaviest elements and dispersing them in space. The nucleus can become an extremely dense neutron star, a black hole or an even more violent gamma-ray burst.

The habitable zone
Let's see again more details about the best candidates, late-F, G and early-K stars (data from here):

The second-to-last column shows the distance a planet should be from its star, for each stellar subclass, to receive the same energy that Earth receives from the Sun, measured in Astronomical Units (the average Earth-Sun distance: about 149 millions km). More generally, the habitable zone is defined as the range of distance from a star when liquid water can exist on the surface of a planet: for the Sun, it's thought to be roughly 120-240 millions km (0.8-1.6 AU). Here there's a calculator that gives the distance needed for early Venus-like, early Mars-like and Earth-like planets, with various degrees of likely greenhouse effect.

Note that considering other sources of heat (greenhouse effect, a stronger radioactive decay in the core) or liquids with different liquidity ranges (see here and here) would allow to extend considerably the habitable zone. Also, planets too cold to have liquid water on the surface might still contain reservoirs of it deeper in the crust, provided they have a hot core.

The habitable zone's width is directly proportional to r2T4, where r is the star's radius and T its temperature: so, if the radius increases 3 times, the habitable zone expands 32 = 9 times; if the temperature raises 2 times, the habitable zone expands 24 = 16 times. Still, when that distance is too close (late-K and M stars) the planet risks tidal locking (also, M stars are believed to be subjected to random and violent flares of UV light), while a planet too far away could not receive enough radiations (solar wind) to shed the primitive hydrogen-helium atmosphere, and become a gas giant (also see here and below).

At this point, we've narrowed our search to metal-rich, main sequence stars late-F to early-K as possible life-bearing stars: roughly the 11% of all the stars in the Milky Way.

Multiple systems
About one third of all the star systems in the Milky Way have two or more stars, though this is more common with giants and hot stars high in the main sequence; the closest to the Sun, Alpha Centauri, is a triple star, made up by a bright G2 star similar to the Sun (Alpha Centauri A), a smaller and dimmer K1 star (Alpha Centauri B) and probably a little, faraway red dwarf (Alpha Centauri C, or Proxima Centauri). Some systems have even more stars.

In such a system, a planet could follow a very convoluted orbit, which would cause wild variations in temperature, and possibly could fling it out of the system, into interstellar space, as often happens with comets. For this reasons, planets are much rarer in multiple systems than they are in single ones. Stable orbits are unlikely anyway in systems with three stars or more, which leaves the two-stars (binary) systems.

The most stable binary systems, taking for granted that all the stars involved are highly-metallic, main-sequence, late-F-to-early-K stars, with a roughly similar mass and luminosity, are those whose stars are either very close of very far from each other: in fact, closer than (0.4√L) AU or farther than (13√L) AU, where L is the stars' luminosity measured in Ls, and more than 30 AU anyway - about one third of the suitable binary systems. This restricts the total number of possibly life-bearing star systems in the Milky Way to 5% of the total, or about ten billion stars.

Anyway, a planet orbiting in a binary system will most probably suffer temperature changes too intense to allow the development of life. This wouldn't necessarily be true if the second star is simply a red dwarf (like Proxima Centauri) or a white dwarf (like Sirius B) at several hundreds or thousands of AU away from the main star.

The planets
Also see: Alien planets

The planets form around the main star and shift until they're placed in a specific mathematical relation: for the Solar System, it's d=(n+4)/10, where d is the distance from the star relative to a reference orbit, the Earth in our case (the distance is therefore measured in AU), while n is a number of the sequence 0, 3, 6, 12, 24, 48, 96, 192, 384, 768, 1536... The reasult for each n is the orbit of a planet, or a large dwarf planet (Ceres, along with the asteroid belt, and Pluto are predicted by the formula).

The energy that a planet receives from its star is inversely propotional to the distance squared: if Planet A is half as far from it than Planet B, it will receive four times the energy for each square meter of its surface (thus it will receive less energy than that if it's smaller, and more if it's bigger). All other things equal, the temperature measured in Kelvin degrees is directly proportional to the energy received: if a planet that receives x amount of energy has a temperature of 150 K (-123 °C), another that receives 2x energy will have a temperature of 300 K (+27 °C). Of course, a thick atmosphere will retain more heat, especially if the gases have a high heat capacity (see the table of atmospheric gases on the linked page).

Orbits and moons
Some specific features of a planet's orbit can be computed through Kepler's laws: Particularly, the third law states that, if we measure the orbital period (T) in terrestrial years and the semi-major axis (d) in AU, T2=d3. As an example, if the planet is 3 AU distant from the star, its year will be long √33 = √27 = ~5.19 terrestrial years (or 1895 24-hour days). Conversely, to have a year of 0.35 terrestrial years (128 days), the semi-major axis of the orbit will be 3√0.352 = 3√0.12 = 0,5 AU.
 * 1) The orbit of a planet around a star is an ellipse with the star in one of its foci (but the ellipse will be likely very close to a circle, and thus both foci will be very close to the centre);
 * 2) A line joining the planet and the star sweeps out equal areas in equal amounts of time (which means that the planet has to move faster when it's closer to the star);
 * 3) The square of the orbital period (the time needed for one complete orbit, i.e. the year) of the planet is directly proportional to the cube of the semi-major axis (half of the ellipse's longer diameter) of the orbit.

With a main sequence star, we can easily infer many parameters of an orbite directly from the stellar mass: (Notes: the Ecosphere is the distance in which the planet irradiation is equal to ± 35% that of the Earth; the orbital period, or year, is measured in terran 24-hour days, and not in local days; the apparent diameter of the star in the planet's sky, assuming a Earth-like distance, is measured both as relative to the Sun's apparent size on Earth and in angular degrees - see below)

Another feature of orbits is eccentricity, the measure of how much it deviates from a perfect circle (which has e=0). For a stable, elliptic orbit it has to be 01 a hyperbolic orbit, both open and not stable). It can be computed for a know orbit as e = (ra-rp)/(ra+rp), where ra is the apoapsis radius (maximum distance of the planet from the star) and rp is the periapsis radius (minimum distance from the star). Currently, Earth's orbit has e = 0.0167, but it can vary from 0.0034 to 0.058 over thousands of years; a great eccentricity can produce wide temperature swings each year, possibly disrupting the climate.

See here a simple simulator of one-planet orbits.

Moons
The length of a "month", that is, the time it takes for a moon to revolve around a planet, can be derived with another simple equation: T = 1.4·√(d3/M), where T is the length of the month in hours, d is the distance between the centre of the planet and the centre of the moon measured in Earth radii (6370 km) and M the mass of the planet in Earth's masses.

For example, let's take the Moon: the distance is 60 times Earth's radius, and the mass of the Earth is obviously 1 Earth mass, so a complete revolution of the Moon around the Earth lasts 1.4·√(603/1) = 1.4·√(216000) = 1.4·465 = 650 hours, or 27.11 24-hour days. Another example: Io is distant from Jupiter 66 Earth radii, and Jupiter's mass is 318 times Earth's mass; therefore, Io revolves around Jupiter in 1.4·√(663/318) = 1.4·√(904) = 1.4·30 = 42 hours.

There is a particular distance, called Roche limit, under which gravitational forces on a part of the moon pull in a different direction than those on another part, thus breaking it apart. Closer to the planet there can be dust rings, but not moons. This minimal distance between the two bodies is d = 2.4·R·3√(DP/DM), where d is still the distance between the two centres measured in Earth radii, R the radius of the planet, DP the planet's density and DM the moon's density. For example, the Roche limit between Earth and Moon will be 2.4·1·3√(5.5/3.3) = 2.4·3√1.7 = 2.4·1.2 = about 2.9 Earth radii, or 18 500 km; that between Io and Jupiter will be 2.4·11·3√(1.3/3.5) = 2.4·11·3√0.38 = 2.4·11·0.72 = about 19 Earth radii, or 122 000 km.

The presence of moons causes tides, by deforming with their gravity the liquid masses on the planet: the magnitude of the tide is directly proportional to the moon's density and to the third power of its apparent size.

Apparents sizes
Knowing the distance and the size of two bodies, of course, we can know the apparent size of one as seen from the other. This size is measured as an angle: the whole firmament is 360° wide. As a reference, both the Sun and the Moon appear roughly half a degree wide in our sky.

The angular size is Θ = arctan 2(r/d), where arctan is the trigonometric arctangent function, r is the radius of the body in the sky and d the distance between the two bodies (they can be in any measure unit, as long as it's the same for both). For example, the Moon has a radius of 1700 km and it's 380 000 km apart from the Earth: let's divide 1600 by 380 000, multiplicate the result times two, and we get 0.0089. As we see here, the arctangent of 0.0089 is about 0.50 degrees. In the same way, Io (with a 1800 km radius, distant 422 000 km from Jupiter) appears 0.49° wide in Jupiter's sky.

Tidal locking
When two bodies, with one orbiting around the other (a moon and its planet, or a planet and its star), find themselves in specific physical conditions, the orbiting body synchronizes its rotation of itself with the orbital revolution, so that it present always the same face to the body it orbits: this is the case for the Moon, of which we can see from Earth only one side.

Given enought time, every pair of bodies will undergo tidal locking. The needed time, measured in Ga (billions of years), is approximately t = (60a6Rμ)/(msmp2), where a is the semi-major axis of the orbit (roughly the distance between the two bodies), R the mean radius of the orbiting body, μ its rigidity (about 3 × 1010 N/m2 for rocky objects and 4 × 109 N/m2 for icy ones), ms the mass of the orbiting body and mp the mass of the central body. All things being equal, tidal locking is faster for closer, larger, elastic (icy) bodies.

For the Moon-Earth system, this formula gives only 3.8 millions years: in fact, the Moon is already tidally locked to Earth, as it was very soon after their formation. Since the time of tidal locking is so strongly related to the distance, planets just a little closer than a certain line (see image on the right) will be already tidally locked at the origin of life, while planets just a little more distant won't lock until the star will be long dead.

Another, simpler formula, which assumes the planet to be exactly identical to Earth and gives the time in years, is t = (6.5 × 104)[a/(0.0273√M)6, where a is still the semi-major axis of the orbit, measured in centimetres, and M is the star's mass measured in grams. For the Earth-Sun system, this gives 1.8 × 1027 years, an enormous timespan: the Sun won't exist anymore long before this time has elapsed.

A world that is tidally locked to its star would experience an extreme disparity between the two sides, especially if the rotation/revolution isn't very fast or the atmosphere isn't very thick: one would be always burning, the other forever dark and icy. In the twilight belt, costant winds and storms would occur where hot and cold air mix (see here for more details). Some real cases of tidally locked extrasolar planets are known, such as Gliese 581-g and Gliese 581-c; while they're generally thought not to be very conductive of life, they could host it in the twilight belt, especially if the locking occurred some billions years after their formation, giving life the time to appear and adapt in a still not-locked world.