The Moon's Orbit

Proposition 35 Problem 16

With a given time, find the inclination of the moom’s orbit to the plane of the eclipltic.

Let:

• AD is the sine of the greatest inclination
• AB is the sine of the least inclination

Bisect BD inC and around the centre C, with the interval BC, describe the circle BGD.

In AC take CE in the same proportion to EB as EB to twice BA. And if to the time given we set off the angle AEG equal to double the distance of the nodes from the quadratures, and upon AD let fall the perpendicular GH, AH will be the sine of the inclination required.

For GE² is equal to GH² + HE² = BHD + HE² = HBD + HE² - BH² = HBD + BE² - 2BH × BE = BE² + 2EC × BH = 2EC × AB + 2EC × BH = 2EC × AH; wherefore since 2EC is given, GE² will be as AH.

Let AEg represent double the distance of the nodes from the quadratures, in a given moment of time after, and the arc Gg, on account of the given angle GEg, will be as the distance GE. But Hh is to Gg as GH to GC, and, therefore, Hh is as the rectangle GH × Gg, or GH × GE, that is, as G H G E × GE², or G H G E × AH; that is, as AH and the sine of the angle AEG conjunctly. If, therefore, in any one case, AH be the sine of inclination, it will increase by the same increments as the sine of inclination doth, by Cor. 3 of the preceding Prop. and therefore will always continue equal to that sine. But when the point G falls upon either point B or D, AH is equal to this sine, and therefore remains always equal thereto. Q.E.D.

In this demonstration I have supposed that the angle BEG, representing double the distance of the nodes from the quadratures, increaseth uniformly; for I cannot descend to every minute circumstance of inequality.

Suppose that BEG is a right angle, and that Gg is in this case the horary increment of double the distance of the nodes from the sun; then, by Cor. 3 of the last Prop. the horary variation of the inclination in the same case will be to 33″ 10‴ 33iv. as the rectangle of AH, the sine of the inclination, into the sine of the right angle BEG, double the distance of the nodes from the sun, to four times the square of the radius; that is, as AH, the sine of the mean inclination, to four times the radius; that is, seeing the mean inclination is about 5° 8½, as its sine 896 to 40000, the quadruple of the radius, or as 224 to 10000. But the whole variation corresponding to BD, the difference of the sines, is to this horary variation as the diameter BD to the arc Gg, that is, conjunctly as the diameter BD to the semi-circumference BGD, and as the time of 20797⁄10 hours, in which the node proceeds from the quadratures to the syzygies, to one hour, that is as 7 to 11, and 20797⁄10 to 1. Wherefore, compounding all these proportions, we shall have the whole variation BD to 33″ 10‴ 33iv. as 224 × 7 × 20797⁄10 to 110000, that is, as 29645 to 1000; and from thence that variation BD will come out 16′ 23½″.

This is the greatest variation of the inclination, abstracting from the situation of the moon in its orbit; for if the nodes are in the syzygies, the inclination suffers no change from the various positions of the moon. But if the nodes are in the quadratures, the inclination is less when the moon is in the syzygies than when it is in the quadratures by a difference of 2′ 43″, as we shewed in Cor. 4 of the preceding Prop.;

The whole mean variation BD, diminished by 1′ 21½″, the half of this excess, becomes 15′ 2", when the moon is in the quadratures; and increased by the same, becomes 17′ 45″ when the moon is in the syzygies. If, therefore, the moon be in the syzygies, the whole variation in the passage of the nodes from the quadratures to the syzygies will be 17′ 45″; and, therefore, if the inclination be 5° 17′ 20″, when the nodes are in the syzygies, it will be 4° 59′ 35″ when the nodes are in the quadratures and the moon in the syzygies. The truth of all which is confirmed by observations.

If the inclination of the orbit should be required when the moon is in the syzygies, and the nodes any where between them and the quadratures, let AB be to AD as the sine of 4° 59′ 35″ to the sine of 5° 17′ 20″, and take the angle AEG equal to double the distance of the nodes from the quadratures; and AH will be the sine of the inclination desired. To this inclination of the orbit the inclination of the same is equal, when the moon is 90° distant from the nodes. In other situations of the moon, this menstrual inequality, to which the variation of the inclination is obnoxious in the calculus of the moon’s latitude, is balanced, and in a manner took off, by the menstrual inequality of the motion of the nodes (as we said before), and therefore may be neglected in the computation of the said latitude.

SCHOLIUM

By these computations of the lunar motions I was willing to shew that by the theory of gravity the motions of the moon could be calculated from their physical causes. By the same theory I moreover found that the annual equation of the mean motion of the moon arises from the various dilatation which the orbit of the moon suffers from the action of the sun according to Cor. 6, Prop. LXVI, Book 1.

The force of this action is greater in the perigeon sun, and dilates the moon’s orbit; in the apogeon sun it is less, and permits the orbit to be again contracted. The moon moves slower in the dilated and faster in the contracted orbit; and the annual equation, by which this inequality is regulated, vanishes in the apogee and perigee of the sun.

In the mean distance of the sun from the earth it arises to about 11′ 50"; in other distances of the sun it is proportional to the equation of the sun’s centre, and is added to the mean motion of the moon, while the earth is passing from its aphelion to its perihelion, and subducted while the earth is in the opposite semi-circle. Taking for the radius of the orbis magnus 1000, and 167⁄8 for the earth’s eccentricity, this equation, when of the greatest magnitude, by the theory of gravity comes out 11′ 49". But the eccentricity of the earth seems to be something greater, and with the eccentricity this equation will be augmented in the same proportion. Suppose the eccentricity 1611⁄12, and the greatest equation will be 11′ 51″.

I found that the apogee and nodes of the moon move faster in the perihelion of the earth, where the force of the sun’s action is greater, than in the aphelion thereof, and that in the reciprocal triplicate proportion of the earth’s distance from the sun; and hence arise annual equations of those motions proportional to the equation of the sun’s centre.

The motion of the sun is in the reciprocal duplicate proportion of the earth’s distance from the sun; and the greatest equation of the centre which this inequality generates is 1° 56′ 20″, corresponding to the abovementioned eccentricity of the sun, 1611⁄12. But if the motion of the sun had been in the reciprocal triplicate proportion of the distance, this inequality would have generated the greatest equation 2° 54′ 30″; and therefore the greatest equations which the inequalities of the motions of the moon’s apogee and nodes do generate are to 2° 54′ 30″ as the mean diurnal motion of the moon’s apogee and the mean diurnal motion of its nodes are to the mean diurnal motion of the sun.

Whence the greatest equation of the mean motion of the apogee comes out 19′ 43", and the greatest equation of the mean motion of the nodes 9′ 24″. The former equation is added, and the latter subducted, while the earth is passing from its perihelion to its aphelion, and contrariwise when the earth is in the opposite semi-circle.

By the theory of gravity I likewise found that the action of the sun upon the moon is something greater when the transverse diameter of the moon’s orbit passeth through the sun than when the same is perpendicular upon the line which joins the earth and the sun; and therefore the moon’s orbit is something larger in the former than in the latter case. And hence arises another equation of the moon’s mean motion, depending upon the situation of the moon’s apogee in respect of the sun, which is in its greatest quantity when the moon’s apogee is in the octants of the sun, and vanishes when the apogee arrives at the quadratures or syzygies; and it is added to the mean motion while the moon’s apogee is passing from the quadrature of the sun to the syzygy, and subducted while the apogee is passing from the syzygy to the quadrature. This equation, which I shall call the semi-annual, when greatest in the octants of the apogee, arises to about 3′ 45″, so far as I could collect from the phænomena: and this is its quantity in the mean distance of the sun from the earth. But it is increased and diminished in the reciprocal triplicate proportion of the sun’s distance, and therefore is nearly 3′ 34″ when that distance is greatest, and 3′ 56″ when least. But when the moon’s apogee is without the octants, it becomes less, and is to its greatest quantity as the sine of double the distance of the moon’s apogee from the nearest syzygy or quadrature to the radius.

By the same theory of gravity, the action of the sun upon the moon is something greater when the line of the moon’s nodes passes through the sun than when it is at right angles with the line which joins the sun and the earth; and hence arises another equation of the moon’s mean motion, which I shall call the second semi-annual; and this is greatest when the nodes are in the octants of the sun, and vanishes when they are in the syzygies or quadratures; and in other positions of the nodes is proportional to the sine of double the distance of either node from the nearest syzygy or quadrature. And it is added to the mean motion of the moon, if the sun is in antecedentia, to the node which is nearest to him, and subducted if in consequentia; and in the octants, where it is of the greatest magnitude, it arises to 47″ in the mean distance of the sun from the earth, as I find from the theory of gravity. In other distances of the sun, this equation, greatest in the octants of the nodes, is reciprocally as the cube of the sun’s distance from the earth; and therefore in the sun’s perigee it comes to about 49″, and in its apogee to about 45″.

By the same theory of gravity, the moon’s apogee goes forward at the greatest rate when it is either in conjunction with or in opposition to the sun, but in its quadratures with the sun it goes backward; and the eccentricity comes, in the former case, to its greatest quantity; in the latter to its least, by Cor. 7, 8, and 9, Prop. LXVI, Book 1.

Those inequalities, by the Corollaries we have named, are very great, and generate the principal which I call the semi-annual equation of the apogee; and this semi- annual equation in its greatest quantity comes to about 12° 18′, as nearly as I could collect from the phænomena. Our countryman, Horrox, was the first who advanced the theory of the moon’s moving in an ellipsis about the earth placed in its lower focus. Dr. Halley improved the notion, by putting the centre of the ellipsis in an epicycle whose centre is uniformly revolved about the earth; and from the motion in this epicycle the mentioned inequalities in the progress and regress of the apogee, and in the quantity of eccentricity, do arise. Suppose the mean distance of the moon from the earth to be divided into 100000 parts, and let T represent the earth, and TC the moon’s mean eccentricity of 5505 such parts.

Produce TC to B, so as CB may be the sine of the greatest semi-annual equation 12° 18′ to the radius TC.

The circle BDA described about the centre C, with the interval CB, will be the epicycle spoken of, in which the centre of the moon’s orbit is placed, and revolved according to the order of the letters BDA. Set off the angle BCD equal to twice the annual argument, or twice the distance of the sun’s true place from the place of the moon’s apogee once equated, and CTD will be the semi-annual equation of the moon’s apogee, and TD the eccentricity of its orbit, tending to the place of the apogee now twice equated. But, having the moon’s mean motion, the place of its apogee, and its eccentricity, as well as the longer axis of its orbit 200000, from these data the true place of the moon in its orbit, together with its distance from the earth, may be determined by the methods commonly known.

In the perihelion of the earth, where the force of the sun is greatest, the centre of the moon’s orbit moves faster about the centre C than in the aphelion, and that in the reciprocal triplicate proportion of the sun’s distance from the earth. But, because the equation of the sun’s centre is included in the annual argument, the centre of the moon’s orbit moves faster in its epicycle BDA, in the reciprocal duplicate proportion of the sun’s distance from the earth.

Therefore, that it may move yet faster in the reciprocal simple proportion of the distance, suppose that from D, the centre of the orbit, a right line DE is drawn, tending towards the moon’s apogee once equated, that is, parallel to TC; and set off the angle EDF equal to the excess of the aforesaid annual argument above the distance of the moon’s apogee from the sun’s perigee in consequentia; or, which comes to the same thing, take the angle CDF equal to the complement of the sun’s true anomaly to 360°.

Let DF be to DC as twice the eccentricity of the orbis magnus to the sun’s mean distance from the earth, and the sun’s mean diurnal motion from the moon’s apogee to the sun’s mean diurnal motion from its own apogee conjunctly, that is, as 337⁄8 to 1000, and 52′ 27″ 16‴ to 59′ 8″ 10‴ conjunctly, or as 3 to 100.

Imagine the centre of the moon’s orbit placed in the point F to be revolved in an epicycle whose centre is D; and radius DF, while the point D moves in the circumference of the circle DABD.

For by this means the centre of the moon’s orbit comes to describe a certain curve line about the centre C, with a velocity which will be almost reciprocally as the cube of the sun’s distance from the earth, as it ought to be.

The calculus of this motion is difficult, but may be rendered more easy by the following approximation. Assuming, as above, the moon’s mean distance from the earth of 100000 parts, and the eccentricity TC of 5505 such parts, the line CB or CD will be found 1172¾, and DF 351⁄5 of those parts; and this line DF at the distance TC subtends the angle at the earth, which the removal of the centre of the orbit from the place D to the place F generates in the motion of this centre;

Double this line DF in a parallel position, at the distance of the upper focus of the moon’s orbit from the earth, subtends at the earth the same angle as DF did before, which that removal generates in the motion of this upper focus; but at the distance of the moon from the earth this double line 2DF at the upper focus, in a parallel position to the first line DF, subtends an angle at the moon, which the said removal generates in the motion of the moon, which angle may be therefore called the second equation of the moon’s centre; and this equation, in the mean distance of the moon from the earth, is nearly as the sine of the angle which that line DF contains with the line drawn from the point F to the moon, and when in its greatest quantity amounts to 2′ 25″.

But the angle which the line DF contains with the line drawn from the point F to the moon is found either by subtracting the angle EDF from the mean anomaly of the moon, or by adding the distance of the moon from the sun to the distance of the moon’s apogee from the apogee of the sun; and as the radius to the sine of the angle thus found, so is 2′ 25″ to the second equation of the centre: to be added, if the forementioned sum be less than a semi-circle; to be subducted, if greater. And from the moon’s place in its orbit thus corrected, its longitude may be found in the syzygies of the luminaries.

The atmosphere of the earth to the height of 35 or 40 miles refracts the sun’s light. This refraction scatters and spreads the light over the earth’s shadow; and the dissipated light near the limits of the shadow dilates the shadow. Upon which account, to the diameter of the shadow, as it comes out by the parallax, I add 1 or 1⅓ minute in lunar eclipses.

But the theory of the moon ought to be examined and proved from the phenomena, first in the syzygies, then in the quadratures, and last of all in the octants; and whoever pleases to undertake the work will find it not amiss to assume the following mean motions of the sun and moon at the Royal Observatory of Greenwich, to the last day of December at noon, anno 1700, O.S. viz. The mean motion of the sun ♑ 20° 43′ 40″, and of its apogee ♋ 7° 44′ 30″; the mean motion of the moon ♒ 15° 21′ 00″; of its apogee, ♊ 8° 20′ 00″; and of its ascending node ♌ 27° 24′ 20″; and the difference of meridians betwixt the Observatory at Greenwich and the Royal Observatory at Paris, 0h.9′ 20″: but the mean motion of the moon and of its apogee are not yet obtained with sufficient accuracy.