Note on the condition for the passage of the Earth through the plane of Saturn's Ring. By H. H. Turner, D.Sc., F.R.S., Savilian Professor. 1. The interesting observations made recently on the ring seen edgewise have brought several inquiries as to the recurrence of this beautiful phenomenon; and the following note, originally made some years ago in consequence of an inquiry from Mr. C. T. Whitmell, may be of use to others. The late Mr. R. A. Proctor gives in his book Saturn and his System a general account of the manner in which the Earth may pass through the plane of the ring, either once or three times, at each favourable opportunity. The present note gives the explanation in more compact form. 2. Let H be the Sun (fig. 1), EGF, ABDC, the orbits of the Earth and Saturn, supposed circular and in one plane. Let BD and AC be two tangents to the Earth's orbit parallel to the plane of Saturn's ring. Then if S be Saturn and E the Earth at a time of passage through the ring, ES must be parallel to BD or AC. Hence Saturn must be either in the portion AB or DC of his orbit. Since his orbit is ten times the size of the Earth's, these favourable opportunities are confined to limited periods which recur at long intervals. AB is about of the semi-orbit, and is described in about a year, so that the Earth meanwhile makes a complete revolution. If, when Saturn is near A, the Earth is near F, then there may be three passages through the ring. 3. To find the condition in exact terms, take HX parallel to AC or BD as axis of x. Let the radius of the Earth's orbit be unity, and that of Saturn's orbit n2. The co-ordinates of the Earth may be written 201 = cos t, y1 = sin t, and of Saturn (remembering Kepler's Third Law) X2 = n2 cos (t − a)/n3, y2 = n2 sin (t − a)/n3, where the unit of time is 1/2 of a year and the origin of time is taken when the Earth lies on HX, Saturn then being an angle a/n3 behind it. The condition for passage of the Earth through the ring is thus 1=2 or sin tn2 sin (ta)/n3, a transcendental equation, which we can only solve by approxi mation. 4. Let us draw the curves (a) y = sint, (b) y=n2 sin(t − a)/n3, then the required values of t will be given by their intersections. Now the curves are both sine curves, differing only in period and amplitude. The Earth's curve (a) alternates much more rapidly, as in ADC (fig. 2). Saturn's curve (b) is a broad sweep, ABC, and we see at once how the crossings of the two curves correspond to the favourable opportunities of fig. 1; and further, that there may be a single intersection as at A, or a triple intersection as at C. There must always be an odd number of intersections; whether there can be five, seven, or more depends on the actual dimensions of the curves. If there is only one intersection, notice that the slopes of the two curves are in opposite directions, i.e. the planets are on opposite sides of the Sun; so that Saturn will be near conjunction with the Sun, and the phenomenon will not be so readily observable. If there are three intersections, the middle one is near opposition and the other two near quadratures. 5. In fig. 3, let ABCDE be two consecutive waves of the Earth's curve, and let FG, HK, LM represent three possible positions of a portion of Saturn's curve, each touching the Earth's curve in one point (F, K, L) and cutting it in another (G, H, M). It is clear that if Saturn's curve cuts the axis OX between T and V there will be only one intersection; if between V and W there will be three. The separating cases are thus defined by the three values of a, OT=α1, OV=α2, OW agi and it is easily seen that = y 6. To find the value of a1, we have the conditions that both Neglecting Since n3 is 29°46, the number of years in which Saturn revolves round the Sun, n=3088, n2=9°54, n = 91*0. quantities of the order n-4 in t, Thus for Saturn-Earth a 1.683, or 96° 25'. Or we can, of course, obtain the numerical values of t and (t-a)/n3 directly from equations (5) and (6). 7. Hence a1=96°, a2=264°, ag=456°. And in the long-run, since the period of Saturn is incommensurable with that of the Earth, the values of a will be distributed uniformly, so that in a long series of years the chance of three intersections is (ag-a2)/360° = 192/360='53, and the chance of one intersection is The intermediate crossing is in the opposite direction and at the opposite point of the orbit. The eccentricity of Saturn's orbit is so large that the intermediate value of a cannot be inferred by elementary considerations. There will be practically two separate series of values of a referring to the two nodes, each increasing by the difference 166°. But this value 166° is only approximate, and is modified by the motion of the ring plane. It appears from the subjoined table that at present the average value is about 162°. 8. The dates when Saturn crosses the line of nodes, or more properly when his ring-plane passes through the Sun, are given by Mr Proctor on p. 223 (Table X.) of his book above mentioned (Saturn and his System). It will suffice here to give a few of them to the nearest day : Columns 1 and 2 above are from Mr. Proctor's book; column 3 is filled in from the Nautical Almanac by finding the date when the Earth had the same heliocentric longitude as Saturn had on the date given in the column before. The next two columns give these dates in fractions of a year, and the differences are the values of a in the unit adopted in § 3, viz. 1/27 of a year, more couveniently expressed in degrees in the next columns. The two series are kept separate, and it will be seen that the common difference is about 162 or 163°. If it were exactly 162° 180° - 18°, the numbers would repeat after twenty terms, and we should have the series as follows: First Series. Year 1789 1819 1848 1878 1907 1937 1966 1996 2025 2055 2084 a No intersectns. 3 I 3 I 3 I 3 3 I 3 I Year Second Series. 1803 1832 1862 1891 1921 1950 1980 2009 α 101° 263° 65° 227° 29° 191° 83° 245° etc. No intersectns. I 3 3 I 3 I 3 I Under the actual conditions, the series will slowly diverge from these, just as the cycle of total solar eclipses on the Earth slowly changes. 9. If the slope of Saturn's curve were less relatively to that of the Earth, there might be five intersections. For observers on Venus or Mercury this might be the case. The limiting case when five intersections are just possible is when a Saturn curve through the point Y (fig. 3), just touches both the adjacent curves near F and K, and then equations (2) are satisfied by a =π. .. from equation (7) n = 2 3π I 5 = 4'54 approx. For values of n greater than 4'54 there will accordingly be three or five intersections, and five will occur with greater frequency as n increases. Now Bode's Law gives for the successive relative values of n2 n2 = 14'3, n = = 3.8 Thus for Saturn-Venus, for Saturn-Mercury, n2 = 25, n = 5'0 Hence Saturn's ring can disappear five times to Mercury, but not to Venus. The value of n for Saturn-Neptune is too small for any but three intersections. It will readily be seen that the above procedure is applicable to the outer planets, the curves interchanging character. 10. Although Saturn's ring is a special problem, there are cases somewhat similar, e.g. the changes in declination of another planet. Times when the declination vanishes are times when the planet passes through the Earth's equator, and thus would be in the plane of its ring if it had one. The three intersections can readily be traced in the tabulated declinations of (say) Venus. |