Some elementary orbital mechanics

Sir Isaac Newton showed that the orbit of a single planet revolving about its star takes the form of an ellipse, with a focus of the ellipse at the center of mass of the system. Since stars are typically much more massive than their planets, the center of mass for most purposes is identical to the center of the star. The elliptical nature of orbits has an important effect on the seasonal cycle, since the planet is farther from its sun at some parts of the year than it is at others. This makes the solar "constant" L a function of time of year. On Earth, we don't notice this effect too much because our orbit is nearly circular. Nonetheless, the effect has an important influence on the long-term evolution of climate. On other planets, it can be even more important.

The distance of closest aproach of a planet to its star is called the perihelion, which we shall call rp. The greatest distance is called the aphelion, which we shall call rap, The semi-major axis is then a = (rp + rap)/2. Let « be the angle made by the line between the star and the planet, defined so that « = 0 at the perihelion. Then, in polar coordinates, the equation of the elliptical orbit is

where e is the eccentricity of the orbit, which lies in the interval [0,1]. e = 0 yields a circular orbit, while the ellipse becomes progressively more elongated as e ^ 1. Specifically, perihelion is (1 — e)a, the aphelion is (1 + e)a and the ratio of the distance at aphelion to the distance at perihelion is (1 + e)/(1 — e). To get the semi-minor axis, we maximize r(K1)s«n(K1), yielding a\/1 — e2. Hence, the ratio of the minor to major axis is %/1 — e2. The geometry of the orbit is summarized in Figure 7.10.

Exercise 7.5.1 What eccentricity would yield an ellipse with a 3:1 axis ratio? Sketch such an Figure 7.10: Geometry of an elliptical orbit with eccentricity e = .66

ellipse, indicating the correct location of the Sun relative to the orbit.

The variation of the solar constant is then given by

where Io is the power output of the star (e.g about 3.8x1026 Watts for the Sun at present). The annual variation in distance from the Sun leads to "distance seasons," which are synchronous between the hemispheres. This contrasts with the "obliquity seasons" (dominant for Earth and Mars) which are out of phase between the hemispheres, one hemisphere enjoying winter while the other suffers under the torrid heat of summer. In the limit of small eccentricity, the ratio of solar constant at aphelion to that at perihelion is 1 + 4e. This represents a very considerable variation, even for modest eccentricity. For the present eccentricity of the Earth (.017), it amounts to 6.8%, or 93W/m2 difference in the solar constant between perihelion and aphelion. To turn this flux into a crude temperature estimate, we divide 4 to account for the averaging over the Earth's surface, and apply a typical terrestrial OLR(T) slope of 2W/(m2K), yielding a temperature difference of more than 11K between perihelion and aphelion. This represents the amplitude of the distance seasons. For Mars, with its present eccentricity of .093, the effect is even greater. The perihelion to aphelion flux variation is 37%, or 219W/m2. For Martian conditions, where the atmosphere has a weak greenhouse effect, this translates into an amplitude of 30K.

To determine the time dependence of r, we must know Ki(t). Because the orbit is no longer circular, the angular velocity is no longer constant; the planet moves faster when is close to the sun than when it is farther away. There is no analytic expression for the time variation of the orbital position. However, it can be easily computed by numerically solving a first order differential equation, which can be derived either from Kepler's equal-area law, or directly from angular momentum conservation. We shall take the latter route. Let v± be the component of velocity perpendicular to the line joining the planet to its star. Then, by conservation of angular momemtum, rv± = J is independant of time. However, the angular velocity of the orbit is simply v±/r, so the angle satisfies the equation dKi = J = J (1 + ecos(Ki))2

This equation shows that the angular velocity of the planet speeds up as it approaches perihelion, and slows down as it approaches aphelion. In consequence, the planet spends less time near the sun than it does at greater distances, and "distance summer" is shorter than "distance winter."

The average of the solar constant over the course of the year can be written

4n a2 r2 r2

where angle brackets denote the average over the planet's year and La is the solar constant evaluated at a distance equal to the semi-major axis of the orbit. We can take advantage of the fact that the same 1/r2 factor appears in Eqn. 7.27 to relate the mean solar constant to the nondi-mensionalized duration of the planet's year. Specifically, integrating Eqn. 7.27 over one year and dividing by the length of the year yields

Ty where t* = Ty/(2na2/J), Ty being the length of the year in dimensional terms. The quantity 2na2/J is the length of year for a circular orbit with radius a. Numerical integration of Eqn. 7.27 shows that the nondimensional year defined in this way decreases as the orbit becomes more eccentric. For e = .1, t* is .995, for e = .25, t* is .968, and e = .5, t* is .866.

Most planets have nearly circular orbits; leaving out Mercury and Pluto, the other planets have current eccentricities ranging from .007 to .093. Even Pluto, the most eccentric planet, only has a value of .244. Note that the difference between perihelion and aphelion distance is O(e), whereas the ratio of major to minor axes deviates from unity by only O(e2). Hence, for small e, the orbit still looks like a circle, but with the Sun displaced from the circle's center by O(e). For small e, Eqn. 7.27 can be solved approximately by a straightforward expansion in e. Substituting

into the equation and matching like terms in e yields the solution k1 = 2nt* +2ecos2nt* + e2 [nt* + y sin4nt*] + O(e3) (7.31)

where t* = tJ/(2na2). The first order term causes an O(e) variation in the orbital angular velocity over the course of the year, but it this term by itself does not alter the length of the year. Taking into account the second order term, it may be inferred that the nondimensional length of the year is approximately t* = 1 — e2/2. In consequence, the annual mean insolation varies very little from what it would be for a circular orbit with radius equal to the semi-major axis. For e = .1, close to the present value for Mars, the eccentricity increases mean insolation by only .5%. For e = .02, similar to Earth at present, the increase is a meager .02%, or .274W/m2. Except in very unusual cases, orbital eccentricity affects the climate through the intermediary of the seasonal cycle, and not through any effect on the annual mean radiation budget.

The consequences of orbital eccentricity for a planet's climate derive from the way the distance seasons interact with the tilt seasons. Each of these types of seasons has a period of one planetary year, so the nature of the interaction is governed by the position in the orbit at which the Northern Hemisphere summer solstice occurs, measured relative to the position of the perihelion. This can be measured by an angle, called the precession angle or precession phase. We will define the phase such that when it is zero, the Northern Hemisphere solstice occurs at the perihelion. It is also common to define the phase as the angle between the perihelion and the Northern Hemisphere spring ("vernal") equinox. When the precession angle is zero, the distance seasons make the Northern Hemisphere seasonal cycle stronger, since "Northern tilt summer" happens when the planet is closest to the Sun and ""Northern tilt winter" happens when the planet is farthers from the Sun. Conversely, the Southern Hemisphere seasonal cycle is attenuated when the precession angle is zero. When the precession angle is 180o, the situation is reversed between the hemispheres, with the Southern Hemisphere getting very hot summers and very cold winters, and the Northern Hemisphere experiencing more moderate seasons. When the precession angle is 90o or 270o, the solstices conditions are no longer modulated by the distance seasons, but instead the vernal equinox becomes warmer than the autumnal equinox, or vice versa.

Figure 7.11 illustrates the effect of eccentricity and precession on the seasonal cycle of insolation. These results were computed by numerically solving Eq. 7.27, and substituting K1(t) into the flux distribution function given by Eq 7.10 and Eq 7.9, after shifting its phase to account for the precession angle. Given K1(t), we also know r(t). Using this, we multiply the flux factor by (a/r(t))2 to account for the variations in orbital distance. This is the quantity plotted, at selected latitudes, in Figure 7.11. One multiplies this flux factor by the solar constant at a distance equal to the semi-major axis, in order to obtain the actual insolation in W/m2. Using the symmetries of Eqn. 7.9, the results for precession angles of 180o and 270o can be obtained from those shown in Figure 7.11 by simply shifting the curves shown by a half year, and interchanging the two hemispheres, so these cases do not require separate discussion.

For both eccentricities, we see that the Northern Hemisphere extratropical seasonal cycle is made more extreme when the precession angle is 0o, while that in the Southern Hemisphere is moderated. At the Equator, the two equinoxes have identical insolation, but the time of maximum equatorial insolation is shifted towards the Northern summer solstice, which is also the time of perihelion in this case. For the larger, Marslike, eccentricity (e = .1), the maximum equatorial insolation in fact occurs at the solstice. For the case of 90o obliquity, the extratropical seasonal cycle has identical strength in both hemispheres, but the equinox conditions now differ from each other, the Autumnal equinox receiving less insolation than the Vernal (Spring) equinox. Also, the time of maximum and minimum extratropical insolation is also significantly displaced from what it would be for a circular orbit. The effect of orbital velocity variations on the seasonal cycle is just barely visible for the lower, Earthlike, eccentricity, but it is prominent for the higher eccentricity case. For 0o precession, Summer is longer than Winter in the Southern hemisphere, while Winter is longer than Summer in the Northern hemisphere; for 90o precession, there is a marked asymmetry between the rate of increase of insolation going into each season, and the rate of decrease coming out of it. For example, in the Northern Hemisphere, Summer sets in rapidly, but the transition to Winter takes a long time. In fact the Northern hemisphere, Southern hemisphere and equatorial insolation maxima are all bunched up within a period of about a quarter of a year, indicating that the distance seasons are beginning to dominate the tilt seasons even at this modest eccentricity. The effect of precession phase on the annual average insolation at each latitude is insignificant; for both the high and low eccentricity cases shown in Figure 7.11, changing the precession phase leaves the annual mean flux factor unchanged to at least four decimal places.

Note that the precession angle has a big effect on climate when the eccentricity is large, but has no effect when the eccentricity is zero. The effects of precession angle and orbital eccentricity work in conjunction with each other, and cannot be disentangled.

At present, Earth's precession angle is close to 180o, so that the Southern hemisphere is driven towards hotter summers and colder winters, while the Northern hemisphere is driven towards a weaker seasonal cycle. This pattern is not manifest in the observations (Figure 7.2) because the Northern Hemisphere has more land than the Southern Hemisphere, giving it a stronger seasonal Obliquity= 20 deg, eccentricity = .1, precession = 0 deg
 Eq 45S / N \ y \ •
 * i / ✓ \ v * X * \i 1 y' #x. * x^ 45N % -

Figure 7.11: The seasonal cycle of solar flux factor for a planet with 20o obliquity, at the Equator, 45N and 45S. To obtain the insolation at any given time of year, this flux factor is multiplied by the solar constant at the time of perihelion. Results are shown for an Earthlike eccentricity of .02 (top row), and a Marslike eccentricity of .1 (bottom row). The left column gives results for a precessional phase of zero degrees, while the right gives results for 90 degrees, both measured relative to the Northern Hemisphere summer solstice.

cycle, owing to its lower thermal inertia. Relatively speaking, though, the Northern Hemisphere seasonal cycle is weaker than it would be if the precession angle were 90o or 0o. Coincidentally, the precession angle of Mars is also about 180o at present, so that the Southern Hemisphere Martian winters are expected to be considerably colder than those in the North. Evidence that this indeed occurs, and its broader implications for Martian climate, will be taken up in Section 7.7.

The precession angles and orbital eccentricities of Earth and Mars have been different in the past, and will be different in the future. This has some extremely important implications for the evolution of climate, to which we now turn our attention.