Imatges de pàgina
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609* From the above value of the radius vector when the equation of the centre is the greatest, the corresponding, true, and eccentric anomalies may be computed by the general equations for those purposes given above. a.1-e 1-ecos.u

·1 pa. 1+ e. cos u;

613. The time from r to S the time from r to Bthe time from B to A-the time from S to A; or, time from B to A time from r to S time from S to A-time from r to B. Now the first of these differences is known being the difference between half an anoma listic year (the time from the sun's leaving the apogee till his return to it) and the observed interval; and the second term of the second difference may be expressed by means of the first. For let the first termt, then the time from r to area r E B TBX EB B=t.

=t

Viz. p= and hence, too, the mean anomaly nt is deterarea SEA SAX EA mined from ntu + e. sine u, and finally being supposed near the apsides) there results the greatest equation of the centre

-nt.

We proceed now to the principles of the method by which the place and motion of the aphelia are determined.

610. It is evident that the sun being in perigee at the least distance and in apogee at his greatest, if we could measure his diameter with sufficient nicety, so as to determine when it is greatest or least, the corresponding places of the sun would be those of the perigee and the apogee respectively; or if, by observing the sun's place from day to day, we could ascertain the times when his angular motion was the greatest or least, his places at the corresponding time would be those of the required points. And if, at a period considerably distant, like observations were repeated, a comparison of the results would shew whether the place of the apogee was stationary or not.

611. Now by the observations of various astromers, it has been found that the apogee of the earth's orbit is progressive, as may be seen from the following statement:

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The mean result of these observations gives about 1′ 34′′ for the annual progressive motion of the apogee of the earth's orbit.

=

612. The following, however, is a more accurate method of determining the progression of the apogee. Let S Er (fig. 13. PlateVI.) be a right line, and draw T Et, making with A B, the major axis, an angle TEA SEA; now the time through rBtS is less than the time through the remaining are SATr; for the equal and similar areas S Et, TEr, are described in equal times, but the area r Et, is less than SET, and it will therefore be described in less time; whencer Et+S Et, which is equal to S Ert S, is described in less time than SET+TEr, which compose the area SETS. This property belongs to every line drawn through E, except AB, the major axis, or the line which joins the aphelion and perihelion of the orbits. Hence, if on comparing two observations of the sun in opposite longitudes, as at S and r, it appears that the time elapsed is not half a year, we may be sure that the sun has not been observed in apogee or perigee. In practice, however, the interval will not differ much from half a year, and the true position of the apogee may be determined in the following manner:

=t.

r B EA E B2 EB SA

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and S

E B2

=

X EAtx EA angular velocity at A r B SA For = tx angular velocity at B EB AE each representing the incremental angle r E B. 614. Now the angular velocities at A and B, or the increments of the sun's longitude, being known from observation, and the time from r to B being expressed in terms of those velocities and of t, the quantity t may be readily determined; whence the exact time when the sun is at A; and his longitude, computed for that time, is the longitude of the apogee.

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Therefore at the second observation, June 30th, the sun was past S. In order to find when he was at S, that is, when the difference of the longitude was six signs (or supposing the perigee to have progressed through 31") when the difference of the longitudes was 6s 0° 0′ 31′′, we must find the time of describing 21′ 49′′ — 31′′, or 21′ 18′′. This is easily effected by this proportion, as the sun's daily motion on June 30th (57′ 12′′): 24 hours :: 21′ 18′′; 8h. 56m. 13s., which taken from June 30th, Oh. 3m., leaves June 29th 15h. 6m. 47s. for the time when the difference of the sun's longitudes under the given circumstances was 180° 0′ 31′′.

615. The interval between this last time and

Dec. 30th, Oh. 3m. 7s. the time of the past observation, is 182d. 15h. 3m. 40s., nearly the time from r to S: but this time is less than half an anomalistic year, which is 182d. 15h. 7m. 1 s., as has been found by repeated observations, and as we have seen above: t-time from r to B 3m. 21s.; and time from r to B = 1. 57′ 12′′

61′

19′′; whence, by substitution and reduction, we have t 47m. 54s. This added to June 29th, 15 h. 6m. 47s., when the sun was at S, gives June 29th 15h. 610. 47s. for the time when he was in apogee.

616. The sun's longitude at that time must be less than his longitude on June 30th, Oh. 3m. by the difference due on the difference of the times, which is 8h. 8m. 19s. This quantity is easily found by proportion to be 19. 21", and

hence the longitude of the apogee is 98°. 31′ by another at an interval of time t, to be a' north 40-5′, or 8′′ 31′ 40-5′′ past the summer solstice.

617. From the longitude at any given time and the annual progression, the position of the apogee and of the axis of the solar ellipse may be found by proportion for any other time. If it were required, for example, to find when the axis of the solar ellipse was perpendicular to the line of the equinoxes, or when the longitude of the perigee was 270°. Now its longitude in 1750 was 9° 8° 37′ 28′′, hence, taking the an.

8° 37′ 28′′

•62

about 500

nual progression at 62′′,years, as the major axis was perpendicular to the line of the equinoxes in 1250. It is remarkable, that the period in which the major axis coincided with the time of the equinoxes, is at the time which astronomers consider to be that of the beginning of the earth.

618. Our next object is to explain those observations made at the earth, and reduced to what they would have been if the observer had been at the sun; as the methods of extricating from the geocentric observations of a planet's place, the elements of the orbit which it describes round the sun.

619. The observations made on the earth are, generally speaking, for right ascensions by the transit instruments, and polar distances by the quadrant or axle The latitudes and longitudes are not observed, but computed from the right ascensions and declinations. Let A, fig. 15. plate VI., represent the first point of Aries, A C a portion of the ecliptic, AB a portion of the equator, S a star, S B its declination, and SC its Latitude; then AB will be its right ascension and AC its longitude round BAC, the obliquity of the ecliptic. Now the method of finding BAC has already been shown; BS is determined by the circle or quadrant, and A B by the time shown by the siderial clock when the sun is on the meridian. Hence AC, CS, and the angle BAC, are given to find A B and B S. Now cos. AS cos AC. cos. CS; cos. SAC

cos. CS. in AC, SAB=SAC=BAC; tan AB cos. SAB. tan. AS and sin. BS sin. A S. in BAS. Hence the geocentric latitudes and longitudes may be always determined. 620. If a the right ascension, d the declination, the latitude, the longitude, and othe obliquity of the ecliptic, then and A may be determined from the following equations:

tan. Asin. o . tan. d. sec. a + tan. a. COS. O sin. I sin d. cos. a sin. a. cos. d. sin. e. 621. By either of these methods the geocentric latitude and longitude may be determined. Among the resulting values of the latitude, some will be either nothing or very small. If the geocentric latitude is nothing, the heliocentric Latitude is also nothing, or the planet is in the plane of the earth's orbit, or in that point of its own orbit which is called its node; the node being the intersection of the orbit of a planet with the plane of the ecliptic. It is not likely, however, that the planet will be observed exactly in the node; but if by one observation its latitude is found to be a south, and

the

at a + a

is the interval, which added to the time when the planet had the latitude, a will give the instant at which it was in its node. 622. As we can thus find the time of a planet's entering its node, we can determine the time of its passage from the descending to the ascending node, and also the time between two successive the nodes and the dimensions and line of returns to the same node; and if the place of the oroit remain unchanged, the latter interval must be the periodic time of the planet; and if the former interval were half the latter, it would prove either that the orbit of the planet was circular, or, if elliptical, that its major axis coincided with the positions of the nodes.

π

623. Now let N P, fig. 14, plate VI. be part of the orbit of a superior planet, NC a portion of the ecliptic, E the earth, S the sun; and let P be an arc of a great circle from P perpendicular to the ecliptic. A spectator at E, sees P under the angle PE, which is therefore the geocentric latitude; and a spectator, as S, would see P under the angle PST, which is therefore the heliocentric latitude. If y be the first point of Aries, then as the diameter of the earth's orbit subtends no sensible angle at the fixed stars, a line drawn from E toy may be considered as parallel to a line drawn from S to y.

Hence

the geocentric longitude of P (L) is Ey the heliocentric longitude of P (P) is Sy the longitude of the sun, (O) is / S Ey, and consequently,

LO+2 SET+E, E representing the angle S E, called the angle of elongation.

624. The angle EST, is called the angle of commutation, (C) the angle S E, or rather the angle SP E, under which the earth's radius appears from the planet, is called the annual pa rallax.

625. To proceed. YST, (P) SE 180°-EST= +180°C, whence P may be determined, if C be previously known. But SE is known from the solar theory, and S E π, or ELO is known, since L can be computed as we have shown above from the observed right ascension and declination, and is known from the solar theory; therefore to find the angle ESπ, and all the other parts of the triangle, it is only necessary to know Sπ, which is called the curtate distance.

626. Now Sπ = SP cos. PST: = 1. cos. H; whence to find S we must know the values of y and H. Let IPN π, represent the inclination of the planet's orbit, to the plane of the ecliptic. Then by spherics, tan. H cos. N. tan. I, whence to find H we must previously know I and N, the distance of the reduced place of the planet from the node of its orbit, which distance is evidently equal to the longitude of the planet, minus the longitude of the node.

627. If the eccentricity of the orbit be small, SP, or r, may be determined by Kepler's law, but it is the mean distance which is determined

by that law; and therefore except P move in a circle, S P so determined will not be quite correct. And in fact there is no direct and general method of determining S P. Astronomers therefore select those positions of a planet in which its heliocentric longitude is exactly known. Now when the inferior planets are in conjunction, their longitudes are exactly known, as when they are in superior conjunction their longitudes are equal to, and in inferior equal to 180° + .

628. In such positions then the heliocentric longitude is obtained without any knowledge of S P, and without trigonometrical computation. The geocentric longitude may be computed from the right ascension and declination by the formulæ already given.

629. If we conceive NC to represent the earth's orbit, and e E that of an inferior planet, then ES is called the planet's angle of elongation, and π E S its annual parallax, when π E is a tangent to E e.

To find the periodic time, mean motion, and dis

tance of a planet.

630. From the observed right ascensions and declinations compute its geocentric latitude; and find when it is equal to nothing. The planet is then in its node. Find in the same way at some subsequent period when it returns to the same node, and thence the periodic time may be determined.

631. This method of finding the periodic time serves also to show whether the orbit is eccentric, and the degree of the eccentricity; as will appear from the following detail given by Delambre, for finding the periodic time of Mars:-July 23d, 1807. in his descending node (88) and his southern latitude increased till Dec. 16th. If the latter time be assumed as that when his latitude was greatest, and the interval (145 days) of his passage from the node to that position, be taken as one-fourth of his periodic time, the period will then be 560 days.

632. But on May 21st, 1808, & in his ascending node (8) and the interval in his passage from 2 to 8 was 302 days. If that interval were half the period, the period would be 604 days. 633. Again on March 7th, 1809, the north latitude of Mars was 2° 49′; and in June 8th, it was (0), when Mars had returned to the node in

which he was on July 23d, 1807, in 687 days, which must be very nearly the period of his revolution.

634 Now from this detail, and what we have done before, we may infer that the orbit of Mars is not circular, and that the major axis is neither perpendicular to, nor coincident with, the line of

the node.

635. But we may draw farther inferences. The time from to being less than the other half of the period by 83 days; if (plate VI. fig. 13) N n represent the line of the nodes, we have = since the areas are propor

NAN-NBn 83

NAn

385

tional to the times. Now when Nn is perpendicular to A B, the difference between N An and NBn is a maximum. In such a position would be nearly

AEN NEB

AEN

41

193

or the

time from B to N would be nearly 152 days and from N to A 193 days.

636. But the period being nearly 687 days in which the planet describes 360° the time of describing 90° would be nearly 171 days, supposing the planet to depart from B, and to move with its mean motion; but as we have seen, the planet was in N nineteen days previously, in which time its mean motion is equal to nearly 10°. When the real planet therefore was at N, the fictitious body moving with the planet's mean motion would be nearly 10° behind. Now this difference is what has been denominated the equation of the centre, which at N is nearly at its greatest value. Hence the greatest equation

of the centre in Mars cannot be less than 10°. The same process for finding the periodic time, and like inferences respecting the eccentricity are applicable to Jupiter and Saturn. But the Georgian planet has not completed more than half a revolution since it was first discovered, and yet we have the elements of its orbit to a very considerable degree of exactness. The following method of determination by La Lande (one indeed of trial and conjecture, but which after a few times is sure of succeeding) will be easily understood.

637. Resuming the notation already employed; the angle of elongation (E)= L-O, L being the geocentric longitude, and ES, the angle of parallax () is the difference of the heliocentric and geocentric longitudes, and therefore equal to P-L. Now EL-is known, and π is

with π

t" and ""-t'.

SE

known from the expression sin. π = sin. E,- Sπ if we can find Sπ. If we assume a value (r) for S (S and SP being nearly equal) we shall from the above equation have a corresponding value of π, and thence of P: let this value be represented by P. Make another computation and a second and third geocentric longitude, and let the resulting heliocentric longitudes p", and PP', and from the three times of be P" and P". Then we have P” — P′, P′′”. observation, t; t" and t" we have t” — t′, t””. Hence P-P': ""-t':: 360°: planet's period, P-P' : t"-:: 360°: planet's period, As p". · P" : t"" —t" :: 360°: planet's period. the period be computed; but r is assumed as the 638. By any of these three proportions may mean distance, and if 1 the earth's mean distance, and p its periodic time; the periodic time of the planet will be represented by pr; and if this result agree with the former one, it will be a proof that r has been rightly assumed; and the disagreement by its nature and magnitude will point out the manner and extent of correcting the first assumption for y.

Or

observations of the planet made on April 25th, 639. La Lande computed from three geocentric July 31st, and Dec. 12th, 1781, and he found from the above formula, the periodic time. The two values disagreeing he amended his first assumption, guided partly by conjecture and partly by his first trial, till a value of r was obtained. which agreed with all the observations.

640. The u stance of an inferior planet may also be deter. ued from observwns on its dis

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mean distance.

an equation from which I may be obtained, either by approximation, or the solution of a quadratic equation.

644. The next step in the investigation is the determination of the form of the planetary orbits. For the sake of simplifying the problem, in the first instance, we shall suppose that the planet's orbit lies in the plane of the ecliptic. Since the mean motion is known from the periodic time, and by observing in opposition or conjunction the planet's true longitude we can at any instant Then if the determine its mean longitude. elapsed time were the interval between two conjunctions, and the orbit were circular, the computed mean longitude would agree with the last observed longitude; and a difference would be difference must depend both on the eccentricity an indication of the orbit's eccentricity; which and the place of the aphelion.

ject in hand, let N (fig. 14, plate VI.) be the 645. To apply these considerations to the subnode of the orbit. Then as its longitude may be considered (from what has preceded) as known, and the longitude of a planet when in eonjunction with the sun is known, being equal to 180°

641. We proceed now to the method of determining the place of the node of a planet's orbit, and the inclination of its orbit to the plane of the ecliptic. In fig. 16, plate VI. let N n, represent the nodes. Now from the observed right ascension and declination we can in an hour even compute the planet's geocentric latitude, and when this is equal to 0, the planet is in its node. Lat. E, E be the two positions of the planet when, as viewed from the earth, it is respectively at n and N. Then S En geocentric longitude of planet at n and SE'NO'-geocentric longitude of planet at N. Now we already know how to compute S N or Sn, and hence in the triangles S En, SEN, we can compute the O, if we deduct the longitude of the planet angles n SE, Sn E, and NSE, SNE'; and from the longitude of the node, there remains thence heliocentric lon. of n=180+―ZnSE and heliocentric lon. of N = CO-180+ NSE', and 'representing the sun's longitudes at the two times of observation; and the angle ES E' is proportional to the earth's motion during the planet's passage from n to N.

642. It is evident that the determination of the place of the node is the more difficult, the less is the inclination of the planet's orbit; and it is difficult on this account to determine the nodes of the orbits of Jupiter and the Georgian planets. 643. The longitude of the node being found, the inclination of the orbit may be thus determined: Compute the day on which the sun's longitude will be the same, or nearly the same as the longitude of the node, the earth will then be nearly in the line of the nodes Nn, at some point e, fig. 16, plate VI. On that day observe the planet's right ascension and declination, and thence deduce the geocentric latitude (G.) Then

tán. I.=

tan. G

sin. t Se

sin. Nt

Det tan. G=St tan.G sin. Sep sin. E tan, G; but sin. N t=cot. t Np. t p; or tan. I sin. Nttp (I denoting the inclination); whence A like diagram and a similar sin. E process will apply to a superior planet. The inclination may also be determined from observing the planet at conjunction when its latitude is considerable. If the planet's distance from the sun reduced to the ecliptic, I the inclination, and G, as above, the geocentric latitude. Then it may be easily shown, that :

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tan. G

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2

-r+ tan.

1) sin. (0-8)

N π. Now as the elliptical motion takes place in the orbit N P it is requisite to know N P, and other like distances of the planet from its node. But N being known, and the angle PN; the distance N P may be computed. For let P N= cos. N. cos. Nπ.

646. If we set off on the orbit of the planet an arc (A) = N r, the longitude of the node, we shall have A+ N P which is called the longitude of the planet on its orbit; and we can have as many such longitudes as there are observations in conjunction or opposition.

647. Three observations are sufficient to deter

mine the two elements of the eccentricity, and
the place of the aphelion; for if we have three
longitudes (V', V, V,) we have two indepen-

dent differences of longitude, and as soon as the
planet's period is known, we can compute two
portions of its mean motion corresponding to
the two corresponding noted intervals of time;
and the two real differences of longitude com-
pared according to the elliptic theory, with the
corresponding portions of mean motion, will give
and place of the aphelion.
us two equations for determining the eccentricity

648. Let e be the eccentricity (supposed to be very small) the longitude of the perihelion, the place of which suppose to be at some point between N and P, and let M, M', M", be the mean anomalies reckoned from the perihelion.. Then we have

V− = M + 2e. sin. V
V'— $ = M' + 2 e. sin. V.
V”— ¢ = M”+ 2 e . sin. V”-

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