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
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.
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