Probability That At Least One Planet Orbits Within The Ecosphere Of A Given Star

As discussed earlier, primaries having masses greater than 0.88 solar mass have complete ecospheres, while those with masses between 0.72 and 0.88 solar mass have narrowed ecospheres because of the rotation-retarding effects of the primary at the inner edge of the ecosphere. To estimate the probability that there will be at least one planet orbiting within an ecosphere, it is necessary to know something about the spacing of planets in planetary systems. Unfortunately there is only one planetary system available to us for study, our own solar system. Since the planets in our system are spaced in a manner that conforms with criteria for long-term orbital stability, however, it seems justifiable to assume that other planets are spaced in accordance with the same laws that have operated here. The spacing of the orbits of planets in the solar system may be seen to be compatible with permitted nearly circular orbits in the restricted three-body problem. The only interference is between the orbits of Neptune and Pluto and this is not strictly an interference because of the inclination of Pluto's orbit. The remainder of the solar system is approximately half taken up with planetary orbits in such a manner that there are comfortable gaps between them. It is assumed that this pattern may well be a universal feature of planetary systems around other stars.

Using this assumption based on our solar system, it is now possible to compute the probability that at least one planet orbits in the distance interval Dt to D0. Taking the actual mean orbital distances from the Sun of the planets of the solar system (see Figure 18 on page 51), the question was asked, "What is the probability that a given distance interval, Di to D0, placed at random, will encompass at least one planetary orbit?" For example, for the ratio DjDi equal to 3.41 or greater in the solar system, the interval D{ to D0 can not be placed so that it does not include at least one planet, since the widest gap between planets on a logarithmic scale is between Jupiter at 5.20 astronomical units and Mars at 1.524 astronomical units and the ratio of these distances is 3.41. Hence, for DJDi equal to 3.41, the probability that one or more planets will exist in the interval to D0 is 1.0. For lesser values of D0\Dt, the probabilities are less than 1.0. Complete results for the solar system are given in Figure 36.

Probability

Figure 36. The probability that at least one planet exists in the distance interval Di to D0 (based on solar-system data).

Probability

Figure 36. The probability that at least one planet exists in the distance interval Di to D0 (based on solar-system data).

If we take D0\Di = 1.441 [ = (1.35/0.65)1] for a complete ecosphere between the illumination limits 0.65 and 1.35 solar constants, then it may be seen that the probability of at least one planet orbiting within a complete ecosphere is 0.63. For smaller values of DjDt, as for primaries with narrowed ecospheres in the mass range 0.72 to 0.88, the probability of the existence of one planet is correspondingly reduced. For each type of primary in the mass range of interest, the probability of the existence of one planet is given in Table 13.

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