Planets probably outnumber stars

kw: analysis, extrasolar planets

Microlensing has paid off. This technique is a most powerful method for finding planets of every size about a target star. An international team has reported recent results of several years of searching. This BBC News report summarizes very well the report and many of the implications. One is the bold statement that every star that is not part of a multiple star system is certain to harbor at least one planet. This may be coupled with the discovery by other methods (transits and gravitational perturbations) that at least some multiple star systems also harbor planets. About half the stars are part of multiple systems, so this implies that the minimum number of planets in our Galaxy is greater than half the number of stars in the Galaxy.

The Galaxy is composed of at least 200 billion stars. If, then, there are 100 billion or more planets in the Galaxy, how many of these are similar to Earth? In size, at least, the report cited above claims this number is about 10 billion. This is a very conservative estimate, and I think it it likely that there are many more than this. I base my reasoning on the principle of mediocrity: Our solar system is most likely to be close to average. Can that be quantified?

Let's make a few rough estimates, based on what we know:

1. Our Solar System has 8 planets.
2. It contains at least three bodies, including Earth, that are expected to have large amounts of liquid water over great spans of time: Earth, Mars (for its first 2 billion years), and Europa (under a thick ice layer).
3. There is one Earth, with life and even (somewhat) intelligent life.

I'll use a Poisson Distribution as a model of the likely distribution of the number of planets, and of possible Earths, around other stars. The process is simple: Find the range of mean values that have at least a 50% chance that there are 8 planets per star. Then use those Poisson distributions to glean some measure of the likely range of planetary numbers.

Firstly, we find that if the mean value of a Poisson distribution is 5, the normalized distribution's height at 8 is 0.37, while if the mean is 6, the height at 8 is 0.64, so we'll use 6 as a lower bound. Secondly, if the mean is 12, the height at 8 is 0.57, and if the mean is 13, the height is 0.42, so we'll use 12 as an upper bound. That means that the most likely number of planets, for stars somewhat similar to the Sun, is between 6 and 12, and a further analysis indicates that most such stars will have a number of planets between the "sideboard" values of 3 and 17.

By a star "somewhat similar to the Sun" I mean a star of spectral type F, G or K that is not a member of a multiple star system. That is about 10% of all stars. The "sideboards" above indicate that there are at least 3 planets each, which multiplies out to 60 billion planets in the Galaxy about such stars, with a more likely number of 150-180 billion planets, and the potential for trillion or so.

Among these, how many might have liquid water for at least a couple of billion years? Repeating the process using 3, we get a range of mean values between 2 and 6, with "sideboards" of 1 and 8. Thus there are at least 20 billion planets holding liquid water, and more likely about 60 billion.

Finally, how many sister Earths? When your sample is 1, it is better to use an aggregation technique, and say, suppose that among ten stars, we were to find five sisters to Earth, what could we conclude? A similar analysis shows that a random group of ten stars might have a mean value in the range 4-8, with "sideboards" of 1-11. This works out to a per-star range of 0.4-0.8 with "sideboards" of 0.1-1.1. These seem reasonable. Thus, I conclude sister Earths number at least 8 billion, with 10+ billion even more likely.

Then why, above, did I state that I think there are many more than 10 billion? There are two sources of more Earths. One is the warmer half of the M stars, M0-M5, which outnumber all the F, G and K stars two-to-one. The other is large satellites of super-Jupiters that may be a little outside the habitable zone of the parent star, but who add heat to the satellite by tidal flexing. This is much more speculative, but is not likely to be zero, so it is more probable that there are millions or a few billions of these also (In the Sci-Fi film Avatar the "planet" Pandora is a giant satellite of a super-Jupiter, though you only see this in an early sequence).

I am encouraged that my very rough "mediocrity" estimates are in the same range as that of the scientists who have given this much more thought than I have. The next breakthrough to await is the ability to get a spectrum from an exoplanet. An atmosphere with water and oxygen will fairly shout "LIFE!" to the Universe.

News flash in planetland

kw: citizen science, astronomy, extrasolar planets

OK, here is what planetary transits look like for a very quiet star, a small (~K6) star that does not pulsate or flare as so many of these do. The orbital period is just over ten days, so the planet is close in. It just appeared in my list of "Candidates".

By my calculations, from the amount of light it eclipses, the planet's diameter is 45,000 km, or just over 3.5 times the diameter of Earth. This makes it just a bit smaller than Neptune.

I didn't observe it on the day of discovery, but a few days later. There is one other star for which I am named (along with a dozen others) as having observed it the first day. This is the luck of the draw; the Planet Hunter team's software parcels out light curves randomly to whoever is logged in.

An even more direct look?

kw: astronomy, extrasolar planets

My two prior posts related to the search for life outside the Solar system. The Seti program currently involves listening for radio signals from a civilization on another planet, while the Kepler mission is looking for small planets that eclipse (transit) their host star and are in orbits that keep them just warm enough for liquid water to be stable on the surface. Sometimes we might wonder, "Why not just look for planets directly?" We expect to, some day, but I'll come to that. Let's determine what we are up against.

These days, Jupiter is a bright presence in the evening sky, two or three times as bright as Sirius, the visibly brightest star. How bright would it be if it were far away? More to the point, how bright would Earth be if we were observed by someone about ten parsecs away?

If both Jupiter and Earth were observed from outside the Solar system, Jupiter would be about five times as bright as Earth. Although it is five times as far from the Sun, it is 10.7 times the diameter, or 115 times the area. Now move away ten parsecs (32.6 light years). This is the standard distance for defining absolute magnitude. At this distance, the Sun's brightness is 4.8 magnitudes. At this same distance, the Earth's brightness is at most 27.6 magnitudes. If we calculate out what this means, the Sun is about 1.3 billion times as bright as the Earth.

Modern telescopes can see stars of 30th and even 32d (apparent) magnitude. The kicker is, not when they are very close to brighter stars, particularly not stars of fifth magnitude, which is very bright. From ten parsecs away, the maximum separation between the Earth and the Sun is 0.1" (a tenth of an arc second, or 1/36,000 degree). That is very close indeed. Although a telescope such as the Hubble Space Telescope can resolve objects that are within 0.03" of one another, that is only if their brightness is quite similar. Take a look at the following image.

This is a small portion of the Hubble Deep Field. While the astronomers purposely picked a nearly star-free bit of sky to look through at distant galaxies, they could not avoid a couple of stars. The one near center of this image is about a fifteenth magnitude star. See how light scattering inside the telescope has puffed up its image to a size larger than the image of many of the smaller galaxies nearby? That illustrates the problem.

A point only 0.1" from the center of that star's image is well within the washed-out circle, but a planet would not be visible even in the much larger gray-green halo. What can be done to improve the contrast?

A mission NASA is planning, provisionally called the Terrestrial Planet Finder, would use four telescopes attached to a long boom, whose light could be combined to null the star's bright image, while enhancing that of a very nearby planet. The last I read, there isn't a set launch date, just a fuzzy objective a decade or more away. When you are working against a billion-to-one contrast ratio, over angular distances much less than an arc second, this is the kind of approach that is needed.

May the time come that we can look at a planet directly and measure its potential for life, or the level of life present! It makes me wonder just how many years away we are from a time when we will know for sure that some kind of life exists "out there."

Getting a direct look

kw: astronomy, space science, extrasolar planets, stars

As a pre-requisite to some of the discussion below, it would be a good idea to read this Stellar Classification article.

The Kepler mission to discover earth-like extrasolar planets has been under weigh for most of a year. It occurred to me to do a calculation or two to see what they are up against. This space mission uses a telescope, a smaller version of the Hubble telescope, to look at thousands of stars, searching for planetary eclipses.

Among the questions you want to answer when planning such a mission are

• What kind of star to watch
• How sensitive the detector needs to be
• How frequently each star needs to be checked
• How many stars to watch

There are others, but these are the biggest factors.

What kind of star to watch? The star classification is most important, because it determines how long the star will stably warm a planet without melting it. The sun is a G type star, specifically G2. From heaviest to lightest, and along the Main Sequence (the portion of a star's existence that it is relatively stable), stars are classified O B A F G K M. Heavy stars are rare but very bright, so we see lots of O, B and A stars in the night sky. Light stars, lighter than the Sun, are very abundant but less luminous, so not many such stars are visible to the naked eye. The brightest G type star is the famous Alpha Centauri double star, visible from the southern hemisphere, which is of the first magnitude, appearing about one-tenth the brightness of Sirius, the brightest. It is only 4.4 light years away, so it can be that visible. About a tenth of the total brightness is supplied by Beta Centauri, a K type star that is so close it takes a telescope to distinguish them.

The Sun is a young-to-middle-aged star, just 4.5 billion years old. It has steadily warmed, now being 40% brighter than it was 4 billion years ago. It will continue to do so, and the Earth will become too hot for life in about half a billion to one billion years. So a G2 star can keep a planet habitable for about five billion years. That has been long enough for complex life to arise in this case, but whether this is common or very rare, we're trying to find out (the Kepler mission is part of the effort).

Let's consider a star that is one-quarter as bright as the Sun. Its classification would be K2 or K3. Its mass would be about 0.7 the Sun's, which means its expected stable existence would be about three times as long. It is a good candidate for hosting a life-bearing planet. The planet is also far enough from the star that it won't be tidally locked, which is considered a detriment.

If we were to consider heavier, brighter, hotter stars than the sun, their "useful lifetime" is shorter. Unless life gets going quickly, and complex life is correspondingly "easy", there is little chance for an A or F star to host aliens we could talk to.

Of all Main Sequence stars, G stars comprise 7.5% and K stars comprise 12%. What about M stars? They are all quite a bit dimmer than the Sun, 8% or less of total energy released. While they have spectacular terms of existence (many billions to trillions of years), most are somewhat unstable, the more so as you go from M1 to M9. Many are flare stars and would periodically sterilize any planet in the otherwise "habitable zone". Some M stars may be suitable hosts for life-bearing planets, but even though M stars in total comprise 75% of all stars, few of these are that suitable. So the focus of a mission like Kepler is on G and K stars. Of the seven exoplanets so far found by Kepler (all of them too hot, but a good test of the system) all were found orbiting stars a bit larger than the Sun. Detector sensitivity is part of the story.

How sensitive a detector? The primary issue here is strong linearity and discrimination. When Earth crosses in front of the Sun, it blocks only 0.000084 of the light, a factor of 1/11,800. You need to be able to "see" such a difference clearly. That means each observation must be long enough to gather plenty of photons so statistical noise is much smaller than the signal. Photon statistics follow a Poisson distribution, which has a standard deviation (SD) of the square root of the mean. Gather one million photons, and your scatter is 99% confined to three SD units, or plus/minus three thousand. That is a third of a percent. Go for ten billion: the scatter in readings will be 300,000 counts, or one in 33,333. That is a good level to shoot for.

This has two implications. One is, you need to return to each star you are watching about hourly. If you are watching 3,600 stars, each one gets about one second of photon-gathering time. Of course, the light is being gathered by a large imaging detector, so you can gather many stars' data at once. In the ideal case, the telescope can be pointed to a single area and watch it for up to a year, gathering thousands of millions of stars' light curves almost continuously. But to gather ten billion photons per star, you need an integration time of sufficient length, which could be from a few minutes to an hour, depending on the brightness of the star.

That is the second consideration. Brighter stars can be usefully measured from farther away, but it is the dimmer stars in which we are most interested. This tradeoff also steers our search parameters in the direction of G stars and the brighter half of K series stars. Only a small number of M stars are close enough.

How frequently to check each star? This boils down to, how long does an eclipse last? If someone is watching us with their own Kepler mission, and they are located right on the ecliptic, they will see a 13-hour eclipse each Earth year. For the K3 star described earlier, the longest possible eclipse is 11.5 hours, but it occurs almost twice as frequently. Back to Earth eclipses of Sol: If the earth is viewed such that it is 0.997 of the Sun's radius from a central eclipse, the eclipse will last just one hour. That is near the detection limit, for the Earth will stay in the region of limb darkening, and darken the Sun my a factor closer to 1/20,000.

This means that the longest time lag between observations should be of the order of an hour. A few minutes is better, but this depends on the brightness of the star and the number of photons our telescope gathers for it.

How many stars to watch? How many Earths do there need to be for a single observer to observe just one of them? The geometry works out to 338. Put another way, only one in 338 stars in the heavens is situated close enough to the ecliptic to detect Earth using the eclipse method. For the K3 star? the figure is 200, because the planet is closer to the star. This means the Kepler mission is more likely to find planets in the habitable zone of K stars than for G stars. If you want to find one hundred Earths, you need to observe twenty or thirty thousand stars.

That is close to the stated goal of Kepler's mission, with one caveat: Not all stars are expected to have planets in "Goldilocks" orbits, not too close, not too far, but just right. Some consider that only 5-10% of the target stars actually have the right kind of planet. Some consider it is closer to 50%. By watching 100,000 stars, we have a pretty good chance to refine this number, at the very least.

A few years from now, we ought to have much better statistics on just how many planets there are in the Galaxy that could harbor life. Then it is up to the engineers to put something in orbit with sufficient data-gathering power (a big mirror!, or 2-3 of them) to look for oxygen in a planet's atmosphere, or other signs of life, while fending off the parent star's glare.

Life with a smaller star

kw: musings, astronomy, extrasolar planets

Late yesterday it was announced that extrasolar planet Gliese 581g, the sixth planet discovered circling dwarf star Gliese 581, is in a "Goldilocks" orbit: not too close, not too far away, but just right, smack in the middle of the habitable zone around the star. The planetary particulars:

• Mass: 3x Earth or more (most likely: 4x)
• Distance to Star: 0.146 AU or 14 million miles
• Equilibrium temperature of an airless body: 228K = -45C (A watery atmosphere's greenhouse effect adds about 35C → -10C average, but can range much higher and lower depending on latitude)

The star's particulars:

• Mass: 0.3 Sun
• Diameter: 0.3 Sun
• Surface Temperature: 3200K (Sun is 6500K)
• Stellar type: M3V (Sun is G2V)
  
• Visible brightness: 0.002 Sun (see below)
• Total luminosity: 0.012 Sun (lots of infrared)
  

This image, from this Wikipedia article (recommended reading!), shows how Gliese 581 would look were it in the vicinity of our Sun. However, if there are inhabitants about the new planet, they are almost seven times as close to a star which is 0.3x the diameter of our Sun, so it would appear twice the diameter in their sky, having an angular width of more than one degree.

That is the first thing that would be different about life on this new super-Earth, so-called because it is larger, probably about 1.5x the diameter of Earth. What else would be different, and what would seem the same?

Firstly, just because astronomers call an M star a "red dwarf" doesn't mean they are all that red. An incandescent bulb's filament has a temperature near 2800K, while a carbon arc (think searchlight or old-fashioned movie projector) has a temperature near 3400K. Both of those look pretty white unless you compare them to sunlight on a clear day, then they look a little yellowish. The Sun is the standard of "white" to our eyes because all the eyes on this planet evolved to take maximum advantage of the Sun's light. Similarly, any creatures on the new planet (which I propose naming Goldilocks) will have eyes adapted to take advantage of its light, which will look white to them, even as light from our Sun would appear a little bluish to them.

So, being closer to the star means it covers a lot more sky, about 4.5x compared to the Sun's apparent area. 4.5×0.012 = .054, or 1/18th and 4.5×0.002 = 0.009, or 1/111. The different sensitivity of the eyes of a resident of Goldilocks would cause the apparent brightness to be closer to 1/20th than to 1/100th. While the ambient lighting will be brighter than the inside of an office building, it won't be by much. And what color would that sky appear? It would be blue, even for one of us. The Rayleigh scattering in a clear atmosphere scatters blue light nine times as well as red light. In fact, it takes a very, very cool star, less than 2000K, to have light that would scatter to look whitish or yellowish rather than blue (to us).

One nice thing about an M3 star compared to a redder star of M5-M9: stability. While more variable than the Sun, an M3 star is not a flare star, so it won't periodically blast the planet's surface with x-rays. The star is considered to be about twice the age of the Sun, but its stellar evolution is so much slower that is is probably only a few percent brighter than it was eight or nine billion years ago (the Sun is 40% brighter than four billion years ago). So the stability of any ecosystem on Goldilocks depends on the stability of its orbit, which we know very little about yet.

One consequence of having much less of the incident light in the "energetic" wavelengths that we humans can see is that there is a lot less energy available for photosynthesis of the kinds we know. Both C3 and C4 photosystems use blue photons with energies greater than 2.6 eV and red-orange photons with energies close to 2 eV. They don't need the green ones in between, so the green light is rejected (reflected). There are precious few 2.6 eV photons that reach Goldilocks, so a different system is needed. Plants there might look black; they have to absorb everything effectively. It'll take some interesting chemistry to utilize infrared photons, if it is possible at all.

This means the energy available to drive a biosphere is correspondingly small, maybe 1/50th to 1/20th of the productivity per acre compared to Earth plants. That corresponds to the productivity on the floor of a thick forest or rain forest. Ferns and other dimness-tolerant plants do well enough there, but life runs at a slower pace. The forest canopy is where the action is. Goldilocks will probably have few places where a canopied forest is even possible, being more of a tundra planet.

And all that is if, big IF, there is life there. While the temperature range is right for having liquid water on parts of Goldilocks, it will be some time before we will be able to determine whether there really is any water there.

My speculation? That it is an ocean planet, and may not have any land that emerges to the air. Look at our solar system. There's lots of ice on the moons of the cooler planets (Jupiter and outward). Mars lost most of its water because it is to small, with low gravity. Goldilocks is cooler than Earth, more like Mars, but is quite a bit heavier. It will have lost very little of any water that came its way, or was part of its formation. I do hope there is some permanent, solid ground somewhere on Goldilocks. It is hard to imagine smart dolphins developing effective telescopes or radio transmitters and thus finding out about the rest of the universe. A radio message has been sent their way, to arrive about 2029. If they receive it, we could get a reply in another twenty years, about 2050. Long before that, maybe we'll have a telescope system that can image the planet directly, so we can see whether it has continents or any weather.