An Estimate for N

Barton Paul Levenson

Introduction

The number of civilizations in the Milky Way Galaxy can be estimated by an equation of the form:

                  k
        N  =  Ns  II  x                     (1)
                  i=1  i

where Ns is the number of stars in the galaxy and x(i) a series of k fractional or probability terms. Ignoring terms which will be known to be important in the future but are unsuspected now, the following equation may be useful:

N = Ns Smr Sms Sp Sc Sg Pa Pd Pe Pm Po Pr fl fi fs (2)

where I use S for terms related to stellar astronomy, P for terms related to planetology and f for terms related to life. These are briefly defined as follows:

Estimates for each term are discussed in separate sections below. It should be noted that some terms are much better defined by current knowledge than others. I arbitrarily divide term estimates into three classes:

FIRM estimates are those well established by known astronomical data.

SOFT estimates are guessable through astronomical data but not well established.

WILD GUESSES are self-explanatory.

Number of stars in the galaxy.

Estimates of Ns usually cluster around 300 billion (Bok and Bok 1981). This appears to be a FIRM estimate.

Fraction of stars in the correct mass range.

This depends strongly on picking a suitable mass range. Masses in general vary from about 120 Solar masses at type O3V to 0.06 at M8V (Lang 1992).

The number of each type of star is related to its mass by the Initial Mass Function (IMF), which can be represented by a simple power law:

             p
    N  =  k M                               (3)

Miller and Scalo (1979) suggest that p has the following values in different mass ranges:

    -3.3    above 10 Solar masses
    -2.5    1-10 Solar masses
    -1.4    below 1 Solar mass

What masses are suitable for the primary of an Earthlike planet? Hart (1979) is of the opinion that ultraviolet flux from stars of type earlier than F7 would preclude life ever colonizing the land. This would correspond to an upper mass limit of about 1.2 Solar masses.

The lower limit is trickier, and depends crucially on the debate over the width of "ecospheres." This term, coined by Strughold in 1955, has recently been replaced by "Continuously Habitable Zone" or CHZ (Hart, 1978), since a knowledge of how the star's luminosity changes is crucial to finding its CHZ limits. In brief, a planet past the inner edge of the CHZ will undergo a runaway greenhouse effect and become a carbon-dioxide desert like Venus. A planet past the outer edge will undergo runaway glaciation and freeze over. Neither type of planet will support life, let alone intelligent life. According to the latest estimates (Kasting, 1993; Kasting et al., 1993), the Solar CHZ extends from 0.95 AUs from the Sun to 1.15. From the mass-luminosity relationship and the knowledge that light follows an inverse- square law, we can say that the outer CHZ edge for any star is found at:

                   2
    CHZo  =  1.15 M                         (4)

The lower mass limit comes about because tidal braking of a planet's rotation by its star follows the inverse cube of distance while illumination follows the inverse square. A planet close enough to a small, dim star to get enough light will be forced by tides into rotational lock, as the Moon is to the Earth. Chances are such a planet will not be habitable -- atmosphere will freeze out on the night side and be lost to Jeans escape on the day side.

But how much tidal force is too much? This is a difficult question, since tidal braking on the Earth is affected by such unpredictable factors as the shape and distribution of coastline. At a rough guess, we might say any tidal force up to that experienced by Mercury from the Sun is all right. This is very generous, since Mercury is locked into 2:3 orbital resonance with the Sun by tidal force, and this, too, probably would make any planet uninhabitable. Days would be too hot for life and nights too cold.

The height of mid-ocean tides (h, in meters) can be found from this equation (after Webster, 1925):

                    4      3
    h  =  0.8506 M r  / m a                 (5)

where M is the mass of the body inducing the tide and m the mass of the body affected, r the radius of the body affected and a the distance between them. Note that an Earth-sized planet at Mercury's distance from the Sun would experience even greater tides than Mercury, since they would be acting over a larger planetary radius.

Mercury, having no oceans, cannot have mid-ocean tides, but clearly tides in the rocks have been enough to trap its rotation. The Sun is 332,946 times as massive as the Earth and Mercury only 0.055274 Earth masses. Mercury's radius is 2.439 million meters and its distance from the Sun is 57.89 billion meters. (All figures from Bishop et al., 1989). If it had oceans, Mercury would therefore experience a mid-ocean tide of 0.935 meters (compare this to Earth's 0.14 meters from Solar tides and 0.305 from Lunar tides). Earth, if moved from its present position, would experience this level at 79.44 million meters, or about 0.531 AUs. We can therefore define a "tidal limit distance" at:

                 1/3
    TLD = 0.531 M                           (6)

we find the minimum suitable mass for a star by setting equations (4) and (6) equal to one another and solving for the mass. This turns out to be 0.629 Solar masses, corresponding to a star of about type K6. Our range is thus 1.2 to 0.629 Solar masses. Using the Miller-Scalo IMF, a range of star masses from 120 to 0.06 Solar masses, the suitable range above, and some calculus, it can be shown that stars in the suitable range would make up about 14% of all stars.

This number would be considerably altered by using a different IMF. Using that of Salpeter (1955), for instance, the figure would be 2.4% -- lower by a factor of six. It could be altered by perhaps another factor of two by calculating different upper and lower limits for suitable stars. Thus this is at best a SOFT estimate -- knowable but not very well known.

Fraction of stars on the main sequence.

This is about 90% (Allen 1973, Lang 1992). This is a fairly FIRM estimate.

Fraction of stars with planets.

Since von Weizsa"cker (1944) revived the nebular hypothesis, most astronomers have assumed that any star, in theory, could have a planetary system. More recently, Asimov (1980) has speculated that the 90% of stars in our galaxy which are relatively poor in heavy elements (the "Population II" stars) may have trouble forming terrestrial planets. Gas- giants like Jupiter would form easily, but bodies like the Earth ("terrestrial planets") would form only with difficulty. Since it was impossible until very recently to tell which stars other than the Sun had planets, it is not clear yet whether empirical evidence supports this contention.

However, computer simulations of planetary system formation have been run as experiments by Dole (1970) and by Isaacman and Sagan (1977). The latter tried varying the dust/ gas ratio from the figure of K = 50 used by Dole (i.e., a protoplanetary nebula of 2% dust and 98% gas). Figures of K = 100, 30 and 10 (corresponding to 1%, 3.3% and 10% dust, respectively) all gave similar results. To simulate the protoplanetary nebula of a Population II star, K should be set to 500, which would give a fairly representative figure of 0.2% dust, and this was not done. But from the above results, my tentative guess on this subject would be that metal-poverty, unless very extreme, is unlikely to prevent the formation of terrestrial planets. I will therefore tentatively set Sp at 90%. This is a WILD GUESS, and will remain so until a good sample of nearby planetary systems is available.

If low metallicity really is a problem, Sp could be as low as 10%. It is important to get this figure resolved more accurately.

Fraction of stars which are safe from catastrophic galactic events

It seems fairly certain now that the center of our galaxy (and likely almost all other galaxies) contains a super- massive black hole. Narayan et al. (1995) estimate its mass at about 700,000 times that of the Sun. Although the cores of some galaxies tend to be violent places, this seems to correlate with age -- the quasars, the most violent of all, are the furthest away in space and thus the youngest in time; the Seyfert galaxies and jet galaxies such as M87 are closer, and the nearby galaxies all tend to be fairly quiet. For a while it was feared that our own galactic center might be saturated with hard radiation. The central black hole is surrounded by a disk of accreting material that gives off hard radiation (and visible light) as it is slowly sucked into the black hole. But Narayan et al.'s analysis shows that most of the energy is "advected" and itself goes down the event horizon. There is thus no reason to think that more than a small volume of the galactic core is uninhabitable due to high radiation levels.

Another possible hazard to Earthlike planets is nearby supernovas (Hunt, 1978), and these do happen fairly often. There is no clear evidence that Earth's ecosystem has ever been negatively affected by one, however, and since the Sun has orbited the galactic core some 20 times since its formation, it has had time to pass by several high-mass stars. We might conclude, intuitively, that disruption by a nearby supernova is a fairly rare event, and assign a 90% figure to the probability that a star's planets will be unaffected by large-scale catastrophes. But a more careful model might show this estimate to be flawed. At present, it is a SOFT estimate at best.

Fraction of stars where binary companions don't disrupt planetary orbits

The simulations of Heppenheimer (1974) indicate that very few binaries will have stably orbiting planets. Even in the widely separated 16 Cygni system, the one giant planet found (Cochran and Hatzes, 1996) has a very high eccentricity (e = 0.634), probably due to the influence of the distant companion. The planet swings from 0.6 to 2.8 AUs, and has undoubtedly caused any terrestrial planets in that area -- including any in the CHZ -- to be ejected from the system. Thus probably only the 22% of stars that are single (Abt and Levy 1976) can have stably orbiting planets.

Worse still, about 3% of the stellar systems so far surveyed for large planets have proved to have "hot Jupiters" in orbit, formed in a process that ejects all other planets from the system (Rasio and Ford, 1996; Weidenschilling and Marcari, 1996; Mayor et al., 1997), except possibly for one other gas-giant in a highly eccentric orbit. DuQuennoy and Mayor (1991) estimate that another 10% have brown dwarfs in orbit, and the brown dwarfs recently found orbiting other stars all seem to have highly eccentric orbits, implying that there, too, other planets have been ejected. Thus we are left with only about 87% of the 22% single stars, or 19% of all stars, that can have planets in stable orbits in the CHZ. This seems to be a fairly FIRM estimate.

Probability a planet is of suitable age

The Milky Way Galaxy appears to be about 11 billion years old (Watson, 1997a, b). Thus no star in the galaxy can be older.

Earth is about 4.5-4.7 billion years old (Allen, 1973; Hart, 1978, 1979). We might take a wild guess that a planet must be at least four billion years old for intelligent life to have evolved. The probability that a planet is of the correct age can thus be related to spectral type by the equation:

    Pa  =  (Tms - 4) / Tms                  (7)

where Tms is time on the main sequence for that spectral type (in billions of years) or 11 (billion years), whichever is shorter.

From these figures and the relative numbers of each type of star, the fraction that are four billion years old or older are about 61%. This would be a FIRM estimate if the four billion year figure were not such a wild guess. As is, this is a SOFT estimate at best.

Probability a planet orbits at a suitable distance

Equation (4) gives the outer boundary of the CHZ using Kasting's estimates; e can also set the inner boundary at:

                   2
    CHZi  =  0.95 M                         (8)

Wherever the tidal distance limit (TLD, equation 6) is farther out than the above, this value must be substituted for CHZi. Thus we can set CHZ boundaries for each spectral type. Noting that Solar planets are spaced roughly evenly in logarithmic terms (Dole 1970), each one 1.30 to 2.01 times as far from the Sun as the last, we can then calculate:

    Pd  =  log R / log (2.01)               (9a)

or (to 99.9% accuracy),

    Pd  =  3.3 log R                        (9b)

where R is the spacing ratio, CHZo / CHZi. This gives us Pd for each spectral type, and as with Pa above, we can weight by relative number and find a mean Pd, which turns out to be 24%. (Results for Pa and Pd for each spectral type are summarized in Table 1).

The width of CHZs has been the subject of much discussion. Huang (1959) guessed at a very wide "ecosphere" for the Sun, stretching from 0.5 to 1.5 AUs and including both Mars and Venus as potentially habitable planets. Dole (1964, pp. 66, 84-85) suggested narrower limits at 0.86-1.24 AUs; Hart (1978, 1979) found narrower limits still at 0.95-1.01; Kasting (1993; Kasting et al., 1993) found wide limits again.

This may seem like quite a bit of bouncing, but in fact each method was an improvement on the last. Huang never claimed to be making a precise estimate. Dole used a static climate model based on the Earth. Hart accounted for thermal evolution, noting the possibility of thermal runaways. Kasting accounted for a stabilizing feedback Hart missed. Thus, at this point, I would guess that Kasting's figures are fairly well established, and thus that 24% for Pd is a FIRM estimate.

Probability a planet's orbit is not too eccentric

If a planet's orbit is too elliptical, illumination at perihelion and aphelion will overwhelm the climate effects of axial tilt and the planet will probably be uninhabitable. Dole (1964, pp. 66-67) estimated e = 0.2 as a safe upper limit. From the eccentricities of planets and moons known in the early '60s, he estimated (pp. 93-94) that 94% of planets would have low enough eccentricity. Many more planetary satellites have been found since then. Of the 61 eccentricities for planets, moons and asteroids listed by Hartmann (1983, pp. 472, 474), nine, or 15%, have eccentricities higher than 0.2, for Pe = 85%.

One can dispute about which eccentricities should "count." Some of these figures (e.g. for asteroids) are for bodies subject to greater perturbations than others (e.g. major planets). If major planets only are considered, seven of nine (78%) have low eccentricities and two (Mercury and Pluto) don't -- assuming Pluto counts as a planet. Both Mercury and Pluto are at the extremes of the Solar system, without a major planet on each side to dampen primordial eccentricities through "collisional natural selection" (Dole, 1970; Isaacman and Sagan, 1977). A habitable planet, especially orbiting a late K-type star, might well turn out to be the first planet in the system and thus to have a high eccentricity. I will take the middle estimate, Pe = 85%. This must remain a SOFT estimate until a clear theory of planetary eccentricities is worked out or we are able to survey other planetary systems.

Probability a planet is of suitable mass

A planet in the CHZ of more than about 3.2 Earth masses will retain hydrogen and balloon into an uninhabitable gas-giant (Dole, 1964, p. 48). A planet of less than 0.2 Earth masses will be unable to retain atmospheric oxygen and thus will be uninhabitable (Dole, 1964, p. 54). A slightly larger one might have an atmosphere too thin to support intelligent life. The limits are far from clear.

To make things worse, planets in the Solar system range from 0.0024 Earth masses (Pluto) to 317.9 (Jupiter). How can one guess the probability of a planet being the right mass?

We have some clues from the fact that the inner ones are rocky "terrestrial" planets and the outer ones (except for Pluto) all gas-giants or "Jovians." Beyond about 2-4 AUs, Sunlight from a young Solar-type star will be feeble and planetesimals will contain as much ice as rock (Hartmann, 1983, p. 133). Hydrogen from the protoplanetary nebula will also be retained more easily by a colder world. Thus there is some reason to suspect that planets in the CHZ will usually be terrestrial planets.

Usually but not always. Recently, "hot Jupiters" have been found in close orbit of Sunlike stars (Mayor and Queloz 1995, Mayor et al. 1997). Apparently these are a special case, created not in place but farther out, and moved where they are by gravitational interaction with another jovian that formed too close (Rasio and Ford, 1996; Weidenschilling and Marzari, 1996). Such systems eject all other planets, except possibly for one other jovian in a highly eccentric orbit. Thus they do not have "planets" in the sense required and can be ignored; they are in the 3% listed in section 5 as stars "without planets" in the sense required.

Still, current models of how planetary systems form have at least been made somewhat shaky by the discovery of such odd morphologies. Since our own system has two planets (Venus and Earth) which are almost certainly of the correct mass and eight (counting the asteroids as one) that almost certainly are not, we can guess that perhaps 20% of planets in CHZs will be of the correct mass. This is a WILD GUESS, though by coincidence it is almost identical with Dole's estimate of 19% (1964, p. 92).

Probability a planet's equatorial obliquity is suitable

Dole (1964, pp. 89-90) estimated 54 deg. as an upper limit to axial tilt, basing a probability figure Po = 81% on the obliquities known for major planets at that time. Since then the eccentricities of Venus and Pluto have been discovered, both of which are very high (177 deg. and 97 deg., respectively). Of course, that of Venus can also be treated as 3 deg. but "rotating backwards." For the climatic stability desired, only Uranus (98 deg.) and Pluto present a problem. Thus we can estimate Po = 7/9 or 78%, coincidentally not far from Dole's value. This is a reasonably FIRM estimate.

Probability a planet's rotation is suitable

Even if outside the tidal distance limit, a planet might have rotation tidally slowed by close or massive satellites, as the Earth in fact does, though not enough to make it uninhabitable. By way of contrast, Pluto is locked into a 6.9-day-long 1:1 resonance with its satellite, Charon.

Of the major Solar planets, three (Mercury, Venus and Pluto) are rotating very slowly and all the rest are rotating very quickly, for Pr = 67% (the asteroids must be ignored since they are rotating rapidly from recent collisions). But the situation is more complicated than that. The gas-giants all receive so much tidal force from their satellites that they would have been slowed or stopped had they not been both extremely massive and lacking in a solid surface with coastlines. This would indicate that only Earth and Mars have intrinsically fast rotation (Pr = 22%).

For years it was assumed that the "planetesimals" which aggregate to form planets were very small bodies (Dole, 1970; Isaacman and Sagan, 1977), so that accretion could be treated statistically. Elaborate models were constructed of how fast terrestrials planets would rotate, generally concluding that larger planets would rotate faster and that all the terrestrial planets rotated quickly at first (see, e.g., Dole (1964), pp. 45-46, and Giuli (1968)). Slow-rotating Venus was assumed to have been affected by Solar tidal drag, though it was only slightly closer to the Sun than Earth and had no moon.

But recent opinion has come to favor an intermediate stage with very large planetesimals, some as large as the Moon (Wetherill, 1991) or even Mars (Dones and Tremaine, 1993). To my knowledge, the possibility of large planetesimals was first suggested by Cameron (1973). Dones and Tremaine have shown that Venus most likely achieved its current slow rotation as the result of a late impact in the "wrong" direction by a Mars- sized planetesimal. So even without tidal drag a planet might wind up rotating very slowly.

Dones and Tremaine suggest that the gas-giants spin by accretion, since on their scale even a large planetesimal is a small one, while the terrestrials spin stochastically and cannot be predicted. If we restrict ourselves to the inner planets, we can take Pr = 50% from comparing slow-rotating Mercury and Venus to fast-rotating Earth and Mars.

A clear theory of satellite formation could suggest how likely a planet is to have a large satellite that could slow it critically. Statistics as to how rotation is distributed in planets with and without satellites must await more detailed knowledge of other planetary systems. Till then, the figure found for Pr is likely to remain a WILD GUESS.

Fraction of suitable planets where life has arisen

The consensus among biologists is that, given a suitable early environment, life will always arise (fl = 100%). A detailed mathematical analysis I have made elsewhere (Levenson, 1997) reaffirms this result. This is a FIRM estimate.

Fraction of ecosystems where intelligence has arisen

The studies of the "encephalization quotient" (EQ) by Jerison (1973) and Russell (1981, 1983) show that this quantity has been rising in a more-or-less steady exponential curve for 700 million years. If an ecosystem achieves metazoan life, it should eventually achieve intelligent life as well. Biologists seem to agree that the advent of metazoans is linked to steadily rising levels of atmospheric oxygen (see, e.g., Knoll, 1991), which is inevitable in an ecosystem with photosynthesis. Thus it would seem that fi is also 100%.

But a planet once habitable may not always be. Kasting (1989) suggests that the steady increase of the Sun's luminosity will bring about a runaway greenhouse only 600 million years from now. There would thus seem to be only a narrow "window" between the advent of metazoan life and that of intelligence. If, for whatever reason, intelligence does not arise before the planet becomes uninhabitable, it never will. For that reason, fi will be estimated at 90%. This is a WILD GUESS. Better (more precise) evolutionary biology and astronomy will be needed to pin it down more closely.

Fraction of intelligences that have survived until now

To know this we must know at least two things: first, the probability that an intelligent species does not tend to destroy itself. From our own experience, this is unclear. It has been suggested (Turco et al. 1983) that the detonation of as few as 100 nuclear warheads at once could set off runaway glaciation. Since the United States and Union of Soviet Socialist Republics came close to nuclear war in 1962, it is possible that many intelligent species kill themselves off early.

The other thing we need to know is how long a species that does not kill itself off early survives. Is there an upper limit to the lifetime of a civilization? If so, what is it? We have no way of knowing.

I will arbitrarily set fs at 50%. This will remain a WILD GUESS until better data is available.

Conclusions

This survey finds the following figures:

    Ns = 300,000,000,000

    Smr = 0.14      Pa = 0.61       fl = 1.0
    Sms = 0.9       Pd = 0.24       fi = 0.9
    Sp  = 0.9       Pe = 0.85       fs = 0.5
    Sg  = 0.9       Pm = 0.2
    Sc  = 0.19      Po = 0.78
                    Pr = 0.5

The product of these figures is N = 25 million. Only some one in 12,000 stars has a planet with native intelligent life.

The volume of the galaxy can be approximated as that of a disc 50,000 light-years in radius and 2,000 light-years thick, plus that of a sphere 8,000 light-years in radius (the galactic nucleus), minus the volume where the two overlap (approximated as a disk 8,000 light-years in radius and 2,000 light-years thick). This is a net volume of 17.5 trillion cubic light-years. The average density of advanced civilizations is thus one per 690,000 cubic light-years, and the average distance between them is 88 light-years.

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Table 1. Star Types as Primaries of Earthlike planets

Sp  Mass    NTms     Pa     CHZ (AUs)      R      Pd

F7  1.200   0.634   5.787  0.309  1.368 - 1.656  1.211  0.274
F8  1.150   0.705   6.575  0.392  1.256 - 1.521  1.211  0.274
F9  1.100   0.788   7.513  0.468  1.150 - 1.392  1.211  0.274
G0  1.050   0.885   8.638  0.537  1.047 - 1.268  1.211  0.274
G1  1.024   0.942   9.313  0.571  0.996 - 1.206  1.211  0.274

G2  0.998   1.003  10.060  0.602  0.946 - 1.145  1.211  0.274
G3  0.972   1.041  10.889  0.633  0.898 - 1.087  1.211  0.274
G4  0.946   1.081  11.000  0.636  0.850 - 1.029  1.211  0.274
G5  0.920   1.124  11.000  0.636  0.804 - 0.973  1.211  0.274
G6  0.894   1.170  11.000  0.636  0.759 - 0.919  1.211  0.274

G7  0.868   1.219  11.000  0.636  0.716 - 0.866  1.211  0.274
G8  0.842   1.272  11.000  0.636  0.674 - 0.815  1.211  0.274
G9  0.816   1.329  11.000  0.636  0.633 - 0.766  1.211  0.274
K0  0.790   1.391  11.000  0.636  0.593 - 0.718  1.211  0.274
K1  0.766   1.452  11.000  0.636  0.557 - 0.675  1.211  0.274

K2  0.742   1.519  11.000  0.636  0.523 - 0.633  1.211  0.274
K3  0.718   1.590  11.000  0.636  0.490 - 0.593  1.211  0.274
K4  0.694   1.668  11.000  0.636  0.470 - 0.554  1.178  0.235
K5  0.670   1.752  11.000  0.636  0.465 - 0.516  1.111  0.151
K6  0.638   1.876  11.000  0.636  0.457 - 0.468  1.024  0.034