Fig. 37. c: The effective inclination of the orbit of satellite 11 (Europa) on the equatorial plane of Jupiter, 1875 to 2175.

Fig. 37. d: The effective inclination of the orbit of satellite 111 (Ganymede) on the equatorial plane of Jupiter, 1750 to 2450.

change by 360 degrees. In other words, 0, A, B and C are 'almost-constants'. In a first approximation, and to trace the behavior of the satellite, they may be considered as constants indeed. Of course, the longitude X of the satellite itself varies very much faster. For instance, the revolution period of satellite III is 7.15 days, so for this satellite the quantity X increases by 50 degrees per day.

By means of the classical formula sin (a — b) = sin a cos b — cos a sin b, we may write expression (2) as

or as

Because the quantities 0, A, B and C are almost constant during a short time period (a few weeks, say), so are the two expressions between parentheses. Let us call these expressions P and Q. So we have

where P and Q slowly vary with time. During a period of a few weeks they may be considered as constants.

Actually, expression (3) is of the same form as (1). To see this, we can pose tan ? = Q/P, so that £ too is a 'constant'. We then obtain

Prove this by way of exercise! In formula (2), 0 is the longitude of the ascending node of the orbit of the satellite on the equatorial plane of Jupiter. Therefore, 0°. 18543 is 'the' orbital inclination of the satellite. However, due to the other periodic terms in the expression for the satellite's latitude, this latitude actually varies according to formula (4). Hence, the satellite now no longer has an orbital inclination of 0.18543 degree, but it has an 'effective* inclination equal to

which, of course, depends on the values of fl, A, B and C, and which, just as these quantities, slowly varies with time. Considered over a short period (a few weeks), the satellite moves as if it had an inclination given by (5) instead of 0°. 18543. The latter value, hence the coefficient of sin (X - 0), is called the proper inclination of the orbit. The 'apparent' inclination (5) can be called the effective inclination.

Said in words instead of with mathematics, each of the periodic terms such as a sin (X - A) has as consequence that the satellite actually follows a path inclined by an angle a to the proper orbit which has inclination i (0°. 18S43 in the case of satellite III) on Jupiter's equatorial plane. The resulting effective orbital inclination will have a value somewhere between i + a and i — a, depending on the position of the 'node' of the perturbation term a sin (X - A) with respect to the node N of the proper orbit.

For satellite III, the effective orbital inclination can have the maximum value (in degrees)

0.18543 + 0.09689 + 0.03924 + 0.01608 + some smaller terms, that is 0°20r. The smallest possible effective orbital inclination for this satellite is

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