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University of Iowa Iowa Research Online Theses and Dissertations 1912 Computation of a comet's orbit Orley Hosmer Truman State University of Iowa This work has been identified with a Creative Commons Public Domain Mark 1.0. Material in the public domain. No restrictions on use. This thesis is available at Iowa Research Online: http://ir.uiowa.edu/etd/3598 Recommended Citation Truman, Orley Hosmer. "Computation of a comet's orbit." MS (Master of Science) thesis, State University of Iowa, 1912. http://ir.uiowa.edu/etd/3598. Follow this and additional works at: http://ir.uiowa.edu/etd THE COMPUTATION OF A COMMET'S ORBIT. Thesis presented in part fulfillment of the requirements for the degree of Master of Science, at the State University of Iowa, June 12, 1912, by O. N. Truman. - 1 - The Computation.of a Comet’s Orbit. The computation of a comet’s orbit is described by both Ilewton and. Laplace as "a problem of great difficulty," and. one to which they had been forced to devote muc5i reflection before reaching a solution. But while it is true that tbedis- covery of the methods of work is a problem which might well have taxed their geniu3, the methods once found are not hard to understand. i will, therefore, give in the following pages a course of proceedure which i think is within the comprehension of one with no further mathematics than analytical geometry, and no greater knowledge of astronomy than is contained in an ordinary elementary course. In such an explanation; of course, the building up of the formulae from their base in the law of gravitation must be omitted, but this omission would have to be made on account of lack of space, anyhow. So if the reader is willing to-assume the foundation stones as laid, he may, by the means I shall set forth,build up the superstructure, and compute the orbit of a comet very satisfactoryKy. The proceedure which I shall use is a combination of the work of Olbers, Euler, and Lambert, which I have modified in a few places for the sake of greater clearness and ease of understanding, even at the expense, perhaps, of lightly more lengthy calculation. I think that it is well for a beginner, especially ir he is a geometrically- and not an algebraicallyminded person, to have a method, every step of which, as far as possible, can be represented by something in a drawing and which he can thus preserve in terms of visual images. - 2 As will be explained, in due course, observations of a comet are taken with reference to movable lines, and therefore when a second and third observation are taken, the lines of reference will be different from what they were at the time of the first. So before any progress can be made in computing an orbit a single fixed system of reference lines must be chosen anci all the observations reduced to it. The description of how to do this will, therefore, be first taken up in this paper; that accomplished, I shall assume the nature of the paths 01 comets in space and the lav/s of their motion in those paths, and on that as a basis show how to find the positions end dimensions of the orbits. I will illustrate each step, as closely as practible, by actual numerical computation as I go along, as it seems to me easier for the beginner to comprehend computational methods when thus set forth than when a great mass of for mulae are first deduced and finally applied to a numerical problem. To illustrate the computation, I will work out the orbit of one 01 Brook’s comets. Comet, 3, 1911 as I saw this one myself and it is thus of special interest to me. discovered by Mr. It was rt. Brooks, at Geneva, Hew York, on July 20, 1911, and at the time was only of about the 10th or 11th magnitude, and approximately opposite the sun. It slowly increased in brightness, traveled north until in the latter - 3 - part of August it was less than 4C° from the north pole, and then swept southward towards the sun. In the latter part of October, when I saw it, it was a very satisfactory appearing comet in the eastern shy, with a straight tail at least 20° long, and a head sometimes of the 3d, sometimes of the 2d, magnitude. By that time it was rapidly receding from the earth, and soon passed from the view of the northern hemi sphere, drifting away in its voyage out into space. Vith these pre] iminary remarks the actual astronomical work may be taken up. The place in the sky of a comet or other heavenly body (to refresh the reader's memory by a resume of the usual elementary treatment of the subject) is established by re ference to two great circles on the‘surface of the celestial sphere--the celestial equator, and the ec>iptic. (Por the sake of brevity "equator" in the ensuing pages will mean "celestial equator" unless otherwise stated.) These intersect at two opposite points, ana that one of those points where the sun in his annual motion comes north of the equator is the vernal equinox. 7/e can now in our imagination pa3S through a commet a meridian or great circle perpendicular to the equator and therefore passing through the poles of the equator. Then the angular distance of the comct from the equator, measured along this mediaian, is its declination, designated by <T Thus declination may be either north or south, the former being indicated by 7. or +, the latter by 3 or -. case it may have any value between 0° and 90°, In either . _ 4 The angle between the vernal equinox ana the meridan we have drawn, from the vernal equinox along the equator towards the east to the meridan, is the light ascen sion, denoted by running from (« oC It is usually specified in time, A j 0 0 0 up to 24%&, but for trigonmetrical cal culation must be converted into angular units, e- oaoil-g aooor-' i'i1 ish e& infro-■aaa^ular- vmi£ft, a thing easily accomplished A ° I to by reason of the fact that 24 is equivalent to 360 , 1 15* , I*1 to 151 , and I5 to lb". All this seems very simple, end would be as simple as it seems if the celestial equator and ecliptic were actual visible great circles traced upon the surface of the sky, and remaining fixed in their places from cen tury to century. To determine the place of a newly dis covered comet, say, it would only be necessary either to measure its angular distance from the equator and the ang ular distance of its meridian from the meridian passing through the vernal equinox, or to measure the distance of the comet from two stars whose position was known, or to measure the distance and direction of the comet from one star whose position was known. The first oi these pro- ceedures, which would be carried out by a transit instru ment, would give «c and S directly; and either of the two latter, for whose execution a micrometer would be used, would give< and S after a little calculation in spherical trigonometry. - 5 - But as a matter of fact the celestial equator and ecliptic are neither visible nor fixed, end the deter mination of the data by which the rt. ascen. and dec. of a heavenly body may be found has been, and still is, a problem of the centuries. All that can be said at present is that we have approximate methods; only the appearance or non-appearance of cumulative errors in the course of time will show how close these approximations are. The reasons for this state of affairs are var ious and to give even a passable explanation of them, ana of the theories by which the many changes of position are taken into account, would require more than the entire length of this paper. I must content myself, then,with referring the curious reader to works on theoretical astronomy for such inform- tion, and confine myself to the bare facts needed in the work before me. It has been found, then, in the theories above hinted at, that the ecliptic and equator both have small motions on the surface of the celestial sphere, and these have been broken up into two classes--secular, or those going steadily in one direction over as great a range of time as man’s observation comprehends, and complex perioaic motions, some going through their changes in about 19 years, some in less. The effects of the former motions - 6 - on the place ox a star are called precession , as are also the motions themselves; the latter motions or their effects are called nutatIon. By taking both of these into account by means of formulae which have been produced for the pur pose^, the position of the ecliptic and equator at any time may be found with reference to their positions at any other time, and as a consequence if the position of a star with reference to the equator and ecliptic of any time are known, its position may be found with reference to the equator and ecliptic of any other time. To make these facts more apparent and at the same time lead to further statements, an illustration may be helpful. If a fly sat on the wheel of a wagon which was being driven along the road, the most suitable way to designate the position of that fly with reference to the road would be to state the position of the wagon, With respect to fixed objects, a mile post in the road,sa^, and then that of the fly with respect to the wagon. The motion of the wagon, being continuous, would correspond to precession; that of the wheel, which would move the fly forward-and back, and up and down, and thus would be periodic, would correspond to ‘nutation. That part of the motion of the fly due to the former cause along might be celled the average oT mean motion, and his position at any time as it v/ould have been had no other cause acted his mean position. If to these were combined bis motion due to the motion of the wheel, and his change of position - 7 - clue to that motion, respectively,we should, obtain his true motion and. his true position at any instant. In just the same way in astronomy, we H a v e the true equator and ecliptic at any date, meaning the actual equator and ecliptic, and the mean equator and ecliptic at that date. r:o get from the former to the latter we need only substract the pert of their motions due to nutation, and to go from-the latter to the former we must add. that part. It has been found most suitable to calculate the position of a comet's orbit with respect to the equator and ecliptic at the' beginning of the !,3esselian fictici ous year," and this time is defined as the instant when the "mean" longitude of the sun, referred to the mean ecliptic and mean equinox of the instant, minus 20".47, is equal to 260? The time of beginning or this fictitious year may be found from the llautical Almanac; for 1911, the year with which we are concerned, it is given on P. 504 as Jan 07764, Washington time. This is in astronomical time, and means 0**.236 before noon on Jan. 1 at Washington, ordinary civil time. In view of all that has appeared thus far, then, the true place ox a star at any time during the year is its mean place at the beginning ox that Besselian year, plus the change aue to precession in the intervening' time, plus the correction due to nutation for the date in question. 'But there is a third disturbing influence, - 8 - aberration of light, which must be remarked upon before the complete reauction formulae can be noticed. This phenomenon of aberration depenas upon the fact that light aoes not travel with infinite vel ocity, but only with very high velocity, ana hence as the earth rushes through space it runs to meet the light rays, so to speak, ana they appear to us to come from a airection slightly uinerent from their true one, in a similar way as vertically falling raindrops appear to be falling in a slanting airection to one who is running through them. It has been touna by complex measurement ana cal culation that light rays falling upon the earth in a line perpendicular to the line of the earth's motion appear to us to come from a direction 20.,r4 7 (according to the Paris Conference ) aifferent from their true one, ena of course when they ao not iall perpenaicularly to the line of motion, the variation, or aberration, is less, becoming zero for rays coming irom exactly the airection towara which, or irom which, we are going. Thus is the value of the 20. ''47 noticea in the remarks upon the Besselian year explained; not, however, the reason for its introduction w hicii in that connect ion must be passed over. This aberration of course afreets the apparent position of every star, except, indeed, those exactly in - 9 the line of the earth's motion, ana its influence on right ascene/ion ana aeclinr tion can he found by spherical trig onometry. The calculation, however, like that of presession ana nutation, is too involved to he gone into here, ana so we must content ourselves with complete and final results alone. • To cotne at" once, t o t h e s e , then, it is possible to unite the formulae for nutation, precession ana aberration, ana sepfrate out cer tain quantities which depend only upon the time, ana others which depend only upon the position of the star in question. ata.rs The former quantities are thus the same for all^at any in stant and can he calculated end tabulated in the Nautical Almanac for every day in each year. These combined with the quantities dependant upon the position of the star, which f we must find for ourselves, give the correction to be added to the mean coorainates of the star at the beginning of the Besselian year, to reduce then to those at any other time within that year. Conversely, of course, if the latter coordinates are known, the former can be found by substracting the correction. There are two systems f6 formulae in use for these purposes, the Besselian and the Independent, ana both are found, together with the tables for use with them, in the llautical Almanac. Inasmuch as the Independant star numbers are most convenient for our purposes, these alone will be explained. On p. 526 of the Aautical Almanac for 1911 are found these two formulae - 10 for each day in the year, - 'f is the time from the ■beginning of the Besselian year, or the time from 1911.0, to use the designation usually employed in this connection. The only place where terns f and of a"fixed" appears in our formulae is in the which have to do with the proper motion star, and we will not need these terms at all, s o /f is of no consequence to us, and the terms containing it will be dropped. f, + f The f in the first formula is „ 0 l the tables; oc, and the star at 1911.0, and oc and a are vttQ coordinates of are its coordinates at the date required. The task with which we are concerned is,however, the reverse of that contemplated in the formulae; there e are given oc, ana problem we have , to find oc and S ; in the present and cT , the observed place® of the comet referred to the true equinox and equator of the date, and it would seem as if the only way to obtain formulae for e>ct and S, in terms of these would be to solve the equations for and S9 . This difficult course is made unnecessary by the fact that «c - , and /- o9 are both very small quant ities, and so we may write oc for op, and / for <£ in the trig onometric parts of the formulae with no material error. - 11 - Doing this, and transposing, we find for the formulae we wish (omitting the terms in -f, which we have no use for) With these equations at hand we can now select observations of the comet, referred each to the equator and equinox of its own date, and reduce them to the mean equator and equinox of 1911.C ^or reasons which will appear later three observations must be taken, and the intervals of time between the 1st and 2d and the 2d and 3d should be as nearly equal as possible. The best length of these intervals cannot be presrcibed, but in the ex ample, as will be seen, it is about 5 days. - 12 - - 14 Certain tilings in this list require explanation. "'irst of all, the 2d and 3d of these observations were taken by means of a micrometer, with which the difference of right ascensions and difference of declinations of the comet and some nearby star were measured. These differences are given in the columns marked A0C and ; the quantities in those columns aie a S to be added, (algebraically) to the coordinates of the cor. parison star in order to get 1he coordinates of the comet. The distinguishing number which the observer has given to each comparison star appears in the column marked and by looking up the same number in his list of com parison stars we find the mean place of that star for 1911.0, the beginning o± the Bessel!m star year. In the same list he gives in the column marked "Authority," the designation of the catalogue from whence the place was taken. Tow returning to the observation list, we find in the column TTHed to ap. plT! the combined corrections for precession, nutation, and aberration which must be added to the mean coordinates of each star to get the apparent coordinates at the time of observation. Fin ally, these app. coordinates plus their4«C and A & , respectively, will give the apparent place of the comet at the time of observation. The work of performing the additions ourselves is save'd us, for the observer has already placed the results in the columns rT oC and " c) ejvfo " meaning, of course, apparent ri'v ' ascension and apparent- declination. « - 15 - the same as given in the respective columns. V , A j these refer to a correction of the observations made necessary by a cause not yet touched upon, i. e., parallax. The observer, it will be remembered, saw the comet not from the center of the earth, but from a point on the surface, and it therefore appeared to him in a slightly different position from what is would if he had seen it from the earth’s center. The influence of this displacement on the two coordinates of the comet depends upon the altitude and azimuth of the comet when observed, the position of the observer on the earth, and the distance of the comet. All these things but the lest are known to the observer, and by special tables computed by him l o r hi ticular observatory he can make the correction much easier than we. Thus he he.s calculated it as far as he could, and gives in the column, the logarithm of the proauot of the parallax correction multiplied by the distance of the comet, the latter expressed in units of the radius of the earth’s orbit. Co if in any way, say from a previously computed orbit.we knew the aistance at the time of an observation, all we should have to do woula be to sub; tract its log- - arithm from 16 "y&jjoA - " and we should have the logarithm oi the parallax correction to be applied to eacn coorainate, thecxl correction in seconds ox time, and 6 correction in seconds of arc. These corrections should be taken as - or +, according as the letter n appears after their log arithms or not. in jny computation I shall not use them, as 1 shall compute the orbit without previous knowledge of the distance of the comet. The only remaining puzzling column is that marked TTComp,r in English and Trench, or "Yerg" in Ger man. Two numbers are usually contained in it; for in stance in the 2d, observation, there is"lo, 10'T. This means that the observer made 15 measurements of AoC and 10 of A S. , The only usefulness of these facts is sometimes in ellaborete discussions of the relative value of observations, end even here they are of doubt ful worth. I shall in this work have nothing more to do with them. Y7ith the exception of one remaining preliminary step we are now ready to take the "*<app!T and B S app." which are in each case referred to the equator snd equinox of the date or observation, and distorted by parallax and aberration, and use the formula heretoxore noticed to reduce them to the places referred to the mean equator and equinox of 1911.0. As I have remarked, the parallax error must be allowed to remain uncorrected. The preliminary step mentioned is the reduction 17 of the times of observation to Grenwich time, end. to decimal parts or a day, instead of hra., rnin.; and sec. To cio this we must look up the longitudes of the places from Greenwich in the nautical Almanac, pp. 520 to t>24,add them (Algebraically) to the times of observation, and reduce the result to decimals of a day. Y/e hove for the three observations: Tables are given in some astronomical works by which the reduction to decimals of a day is easily made. 13y anjr means, however, we get The necessity of making these reductions in this place comes from the fact that they are now to be used in interpolating for 0, H, i, g, and h, which are to be used in making the reductions to mean places for 1911.0. A little inconvenience still arises trom the fact that th $ II i g h ,quant it ies, are given in the tables for 'Washington mean midnight, not for Greenwich - mean noon. 18 - " idnight, July 21 " is midnight, July 21-22. So from each observation we must substract 0.21414 day., the longitude of Washington expressed in days,'and then 0.5 day, to reauce the tine to a day which begins on a midnight instead of the previous noon. Both sub. tractions are of course performed at once by suhstracting C.71414 day and we have - 19 - - 20 - - 21 Thus we can easily find the values of the and <S by substraction. If the values ox the corrections obtained for the 2d observation are compared, with those given by the observer.under "Red. to Appl."» it is seen that they agree exactly, and this of course might have been expected from the beginning, for the corrections are small and slowly changing quanitites, and their variation in passing over the few minutes of cxC between the star and. comet would be insignificant as rar as this hind of work is concerned. This might of course be proven more definitely by differentiation 01 the expressions lor oCe-cC- S.-J with respect too: and d and d , respectively, and substituting numerical values in the resulting derivatives, ’but this is not worth the trouble. In most cases it will be found, as above, that the corrections given for star positions, if they are given r.t all, are fully sufficient for the comet, but it is well in any problem to work out one or two corrections by the formula, as has been done rs&l 1y above, to make sure that the discrepancy is^insignicicant. In this case it manitestedly is, and so for the 3d ob servation I shall izse the values +• 2*24 and 44,"1 with no further question, remembering, however, that these are to he substracted from < and to gevoC* end £0 , not added. - 22 - Since in the method, of computing an orbit which I shall use the plane of the ecliptic ana not the plane of the equator is to hs taken as a fundamental plane of reference, these right ascensions and declinations must now he changed into latitudes and longitudes. The standard -^Oimulae "by vvhich this is accomplished are found - 23 - in many estronomioal works, '.’hey depend for their demon stration on spherical trigonometry, and I will simply give them without deriving them. If o£ and / are the right ascension and dec lination of a star, 6 the ecliptic, and the inclination of the equator to X and ^3 the longitude and latitude, respectively, which it is desired to find, we have first to find - 24 - - 2 5 - - 26 - Having now obtained these latitudes and long itudes, we can go no further without some knowledge of the nature of the paths of comets in space, and the form ulae by which their motions in those paths can he predicted. In this matter again I must ask the reader to take the formulae on faith, for while their derivation from the lav/s of motion is most interesting, it would occupt a prohibitive amount of space. it is shown, then, in celestial mechanics that if we neglect the comparatively insignificant attraction of the planets, and pay attention only to that of the sun, every comet moves about the sim in either an ellipse, a parabola, or a hyperbola, having the sun at the locus, if the first, the comet would return in its path at re gular intervals indefinately; if either of the other two, and we should see it but onceyit would go out into space, never to return, ^hc kina of curve depends only upon the velocity of the comet, and the relation is expressed in the following manner. Let gravitation" be the value which thetfconstant of has if we take the sun's mass as the unit of mass, the second as the unit of time, and the semi major axis of the earth's orbit as the unit of length. This value, which proves to be 0 ! 7 I. D o j is generally called the "Gaussian constant of gravi tation." It is such a value that if ,and. are two masses expressed in units of the sun’s mass, and they - 27 - are separated, a distanceyi- expressed in units of the Sun's distance from the earth, the force pulling them f7~ A * /- w towards each other will be f - 'ft --- — -pO T Q P s i Since acceleration --- -— , if — mass ' were let go it .} v;ould fall toward m, with an acceleration at first —~—i z units per sec, per seo„ ana if raL were let go it would toward m with an acceleration at first = _ It is to he noticed that I have here defined the force in the only way that force can be aefined--by telling what it will do— what acceleration it will impart to a given mass. How let A be the distance of a comet from the sun; and v the velocity with which it is moving; if that comet is moving in a hyperbola; if A / its path is a parabola, and i f 0-<A[JL±-t <2T= travels in an ellipse. It therefore appears that of an infinite number of velocities a comet might have, only one would cause it to move exactly in a parabola, ana so the chances are infinity to one that no comet moves in that curve, It is, moreover, known from observation that of all comets whose orbits have been computea, not more than 3 or 4 appearea to be hyperbolas, ana those were not certainly so, so that on this ground alone there is strong reason to believe that all comets are moving in ellipses. It has proven to be a fact, however, that in the cases of all F ig . 1 . - 28 - "but a few tlieir orbits are so nearly parabolas that their performances oan he very well predicted by assuming a parabola as the true form, and in the case of a newly discovered comet this is always done. I shall deal, then, only with a parabolic orbit. Let 3, figure 1, represent the sun, - & parabola in.which a comet is moving, and V the vertex of that parabola. If r is any radius making an angle v with the line S V, we have from analytic geometry A = y Z SY is in all our future work to be called q, the parihelion distance of the comet from the sun, and v is called the anomoly of the comet. Also let T be the time when the comet is at Y , and t the time when it is at G. Then the following equation holds good x. > 2- ^ i. where k is the 4jaassian constant already explained. 3o knowing q, and T, we can find t for any anomoly v, or conversely we could find v for any given time t. The latter process would necessitate the solution of a cubic equation, and to avoid the trouble of this the equation has been multiplied through by 7b, giving rr -y- yr °ables have then been constructed giving II or log L' for every value of v, (by steps of only I 1 or in some cases only 10” from 0 to 180* and *80 when we have v or I. the other can be quickly found. are called barker’s Tables. These The most complete of * Fig. 2. 29 - them is found in Oppolzer's ,rDe determination des orbits des oometes et planete.s.TT and goes by steps of 10", with interpolation tables for intermediate values. yj A- . or to is found to equal log™/ rT use the usual symbol r If. C0 /I? (9.9601277-10) '7 y 7J so as the complete formula for the relation between v and t we have Thus far, then, I have shown how to reduce the observations of a comet to a known plane of reference, and how to find the place of a comet in its orbit at any time. We are now able to proceed towards finding the dimensions and position of trie oruit in space, the ultimate end of our work. For this purpose we shall take the sun S as origin, ana from it draw an x axis in the plane of the ecliptic through the vernal equinox, r j a y axis per pendicular to this and in the plana of the ecliptic, and a z axis perpendicular to the plane of the ecliptic, and let the ellipse represent the earth's orbit. 7e thus have the beginning of tig. 2, which represents space, of course^ a . a . . - 30 - Suppose now that at the 3 times of observation the earth was at "Jf B l and E_ . The positions of these points oan be founa from the Tautical Almanac, as will he seen later. Then if X y and A t are the longitude and latitude of the comet at the first observation, let the lire t Cy he drawn through B. in the plane of the ec liptic, giving angle h, its proper value, and perpendicul arly above or below it, according a s ^ is -+• or -, let S(0tbe drawn^ making angle C; E4 3/ =y3/ • We now have the position of one line of sight to the comet from the earth, and in the same way we can locate the other two, - A and E fC5 . ITow at the instants of each observation (except for a source of slight inaccuracy which will be taken up later) the comet was somewhere on the corresponding line. Suppose that its three places were c, c^ ana c3 . Sinoe a comet moves in a plane which passes through the sun, the first condition these noints must satisfy is that a m IS a U. n7r single plane^cutting the ecliptic on the linel/L/ and making with the ecliptic an angle i, contains the sun and all three of them. IText, a comet moves in a par abola, with the sun as the focus, so it must be possible to draw such a parabola through 0, 0^ 0 in the plane ^-£2. XJfc • Let the vertex be P, which now becomes the perihelion, or point where the comet was nearest the sun.Then P s will have a certain length cj_, the shortest distance from the comet to the sun, or perihelion distance, and will make Avl cO ^ S . 31 - - ■'inally, if I is the time the comet is at P, then if it is to obey the laws of parabolic motion the anomoliec v \ » - tt 2 o/ of Ctc^must satisfy the equations * n ~TT ^ 2- / i— ^ ^ +7 ^ ^ - Tr ^ %, So if we are to find, three points oo^c, which. truly represent the places of the comet, and from them find the orbit, the points only need satisfy all the above requirements. It is apparent, however, that fulfillment of these conditions would demand that the plane of the orbit be placed in some exceptional position, and it would be hopeless to try to locate it there by direct trial. So the most that we have accomplished by this excursion is the discovery of co-j the four unknown quantities, position of the line CL ; besides the which must be determined. In regard to some of these a few additional remarks are de manded. In the first ^lace the comet in the drawing pierces the plane of the ecliptic and descends below it at the point II, so this point is called the descending noae, IT S, or J*!. \y~ is the line of nodes. comet would ascend and the line Sometime,too, the above the plane of the ecliptic again, at some point out on the line 6 called the ascending node. -Q. and this point would be The lineS-Qis of especial import ance, inasmuch as the angle/I. which it makes with the X axis, measured around from position* T toward the east, fully determines its This angle.whioh may be anything from o'to 360* is ' - always called jQ. 32 - , the longitude of the ascending node, and may he plaoed with i, co , q, and T, making five ■unknowns in all. As regards i, if we stand at the ascending node and look towards the sun, it is reckoned upward from the plane of the ecliptic, towards the left, in the same way as angles are ordinarily reckoned. 0° to 180" . It may he anything from If it is between 0* and 00* , it shows that the comet is moving around the sun in the same direction as the earth and the other planets, or that its motion is direct; if between 90* and 180* , the comet moves in the opposite direction, or retrograde. The angle60may be anything from 0* to 360 , and is reckoned fromSH.along the plane of the orbit in the direction of the comet's motion. T is of coxirse the time when the comet is at j P, the nearest point it gets to the sun, or perihelion. As regards q nothing more need be said. Paving thus cleared up the details in regard to the elements of the orbit, we can proceed again towards the determination of them. As has been remarked, U.IX- the attempt to find them by direct trial would be successA ful, so other means must be sought. A suggestion as to the proper path to pursue is gained if it ie observed that once the points c, and c s alone c%) were properly located on the lines E. 0, and E,v 0.o , that is if the distances Ec,/ and E c.3 , which I shall hereafter call A and , were known, all of the - 33 elements are fixed, ana only //oula need to "be calculated. For from the principles of analytical geometry only one plane coula be passed through S, c/ and c5 , and in that plane, only one parabola with 3 as the focus coula be made , C-j, to go through and c,^ . So J a n d q are all deter mined, and by using one of the times of observation, t f say. in the eauation of course being also known from the parabola, T could easily be found, and our task would be done. Eut I have saia "if the points c( and c3 were properly located in the lines processes coula be carried out. in it the requirements; 0, and Z 11 the above This "properly" contains (1) that the parabola passed through c, ana c^ shoula also contain cz , ana (2) that the T ae bermine a from the equation above usea snoula also sauisiy tne equations ana tne question arises, how we could tell without going through the long calculation hinted at above, whether two chosen values for ^0, andwould of the needful conditions. secure the fulfillment Fig. 3. - 34 - - One step towards the solution of this difficulty may suggest itself upon the- reflection that while v. v3 and T are obtainable from given values of A , vt and ^ 3 only at the expense of exceedingly tedious calculation, the distances of the comet from the sun at the two ZitncS, I r, and r. , and the distance X of the 1st and Sd places of the comet from each other, can he found with comparative ease, ^or consider the 1st observation, and refer to fig. 3. - 35 - F i g . 4. - 36 - It is apparent from plane trigonometry that in the triangle S Q/ o3 Pig. 4 , M L - A , 11 - 2. A , A } fa -V jJ In these five equations there appear seven unknown quantities v v3 T q, r; , ?3 , K , ancl it ought to he possible to entirely get rid of the four unknowns v; , r , T, and q, whose oalciili tions from^anQL/^jinvolves so much trouble, leaving only a definite relation which must be satisfied by the quantities r, It is well known, however, that comprehension of the possibility of eliminating unknowns between equations, and the actual accomplishment of it, are two entirely different matters, and in this case one must pick his way exactly right if he is to meet with success. I will jtherefore, give^method of elimination which is found in Watson's Theoretical Astronomy, without attempting to tell how it was ever hit upon. ($ ) - 3 7 - - 38 - - 39 - - 40 - - 41 - This extraordinary result is due to Euler, hut always 'bears the name of "Lambert’s equation", after Johann Heinrich Lambert, who considered similar relations in the case of the ellipse as well as the parabola, and jpro^uced a far shorter but less trustworthy proof of the theorem. Once in possession of it, it would be more within the range Of practicability to find by trial values of /?/ a n d t o which values for r, , r3 and X satisfying Lam bert's equation would correspond, and finally determine whether the parabola passed through c; and c5 would contain the point ct . It is said indeed,that methods based on this principle are among the best for improving an already known approximate orbit, but to find an orbit at first entirely unknown they would again lead to hopelessly long calculation. Relief from this dilemnja is found in the xorincipfe of Olber's and the resulting equation, which provides means for calculating from known quantities a closely (mate approx^ value of the ratio > 80 ‘clia‘fc in our formulae for- finding r a and^which formerly contained^/^ ^ /3t can be introduced instead. Then Only/?, remains to be found by trial so as to give values of x r a n d * which satisfy Lamberts eqiiation, and from there on the problem is merely one of analytic geometry. U It is^ Olber's equation, then, that we mtist next devote our attention. This equation, takes its origin in a principle F ig. 5. - 4 2 - ox orbital motion not heretofore mentioned** tfhe famous law of Kepler,elaborated by llewton, that if a comet moves about the sun in any of the conic sections, its radius vector sweeps over areas proportional to the times consumed in describing them. Fig. 6. - 4 3It is upon the accuracy of this "nearly' that the Giber's method rests, and as ’’nearly” is the more or less exact, the orbit obtained is the more or less correct. We must, however, take the assumption for what it is worth, and I will later discuss some factors which influence its effectiveness. Assuming A--,. -A — . we may draw, in fig. 6. f e/ c , ct c a figure of a portion of a comet orbit T in space, with the sun at 5 , and a portion of the earth’s orbit E IEL* The planes of tie two orbits intersect on the line s n . Let the places of the comet and earth at the three lines of observation be c, ® tC.and E,E,E3 , respectively, and draw the lines of sight E, o, , E^ 0 v, E o , the radii S* c , 3 ot S o t S E , , S Et ,S E 4 , and the chords c; o and E#~} intersecting 3 oxand 3 7^at q and Q, respectively. We now make the Olber’s assumption as well of the segments of the chord of the earth's as of the chord of the comet's orbit c orbit E,E c , and so say Fig. 7. F i g . 8. F i g . 9 . - 4 4 - Fig. 1 0 . -45- Fig. 11. -46 - Fi g. 12. - 47 - Pirst choose the point Q, and draw the trianles Q o' c/ and Q G^Oj , and the lines Q 3, icj and Q oj in their proper positions. Then draw a sphere about Q, as a center, and produce all the lines going otxt from Q till they cut it. Then if we transfer the letters c/ c/ 0/ q and 3 out to the surface or the sphere, we shall have a spherical triangle S c 'c/ with a great circle drawn across it rrom S through o' cutting the side o' c 7at c. To make the subsequent opera tions plainer, it will he necessary to construct a figure, figure 12, in which the system is seen from a different angle, and the lines are in different relative positions. In that figure Q ;S ,and q have the same significance as beiore while c, c and c3 c orrespond to o/ g ^o , in the former drawing; S I is a portion of the ecliptic, and P is its pole. low it is — - that we must find, and it is apparent that its value depends.upon the angles </, a^ ' fit. and the sides S c, and S c3 of the spherical triangle S c/ Tie nrust now find these quantities. Draw the arc of a great circle ? ^ be the vernal equinox on the ecliptic, L, and let ® ..ill then be the longitude of the sun as seen from Q at the time of the second observation, and because Q is on the line join ing the sun and the earth at that observation, will also he the longitude of the sun as seen from the earth at that observation, and can he found from the Uautical Almanac. will he the longitude of the comet at the first observation, because while the comet was seen from EL; in figure 6, and not from „ , the direction of the line of . - 4 8 sight was preserved intact in transferring that line of sight to the new position where it had its base at Q. For the sane reasons A fwill "be the latitude of the comet at that observation. Angle 3 I c/5 is of course equal to O 90 , for ?c, L passes through the pole of the '7e now know two sides S c*C~ (^/ " &) CL^J~c, •(- ecliptic. f of the right triangle 3 I c, and can find the side 5 a f ^ ich I will aall// and the angle SL, or Y/ • spherical trigonometry we have Going now to the triangles c, Sq and ^ c :, we find that they have a common side ~ q, and that their sides have "become known, as likewise their angles 4 I* K - J ”) we have 4rh'Yt* So referring again to spherical trigonometry -49 - - 5 0 - 51 So we must first oaloulate Q>R,, * :,: T & W, "by the formulae above, and. then r;\ rf and rf are easily found for any assumed value of^<P/ . Since the computation of the nine quantities above is rather long, it is necessary to have a check to guard, against mistakes. A good, one is found in the tact that as is easily seen hy multiplying out these expressions. If the values obtained in this way xor X I anu. W agree with those previously found, it is almost certain that not only K.Ii & W, hut the P's s;ana P.'s are correct. “or the computation of P.* , P_ Q./ , etc., we have already found expressions for A A>;'3(; B,; 3/; 0,, hut the X*s Y's and z»s Were only roughly defined on p. 3 4 IS he in™ the coordinates of the earth at the times of observation. As a matter of fact, for the best results they as the must be token .coordinates of the points on the earth from A whence the comet was seen, at the times of observation, and are each made up of two parts, of "or example, X # is the sum x.e. i , the x coordinate of the center of the earth 7 referred to the sun as origin, and x,/;the coordinate of the point of observation referred to the center of the earth as origin and axis parallel to the former ones as 53 axes. Ir the sarae way y, = zjt/ + < jfpl, and so on. X^i (ylt^4/,^ , ^ A j p-re easily obtained as -follows: On p. Ill for each month in the "autical Almanac (p. 112 for July, 1911) are found the logarithm of the radius vector of the earth, the true longitude or the sun,as seen from the earthy referred to the equinox of 1911.0 fin the column A / and the latitude ^/3h of the sun, for Greenwich Liean iloon of every day. If we let he the longitude ana latitude of the earth as seen from the sun at the time of the first observation, say, it is easy to see that Then if R, is the radius vector of the earth we should have from fig. 5, F ig . 1 3 . -54Similarly we can find, values for The case of *>,, is more com plex, for these depend upon the distance or the place of observation from the center of the earth, the geocentric latitude, and the sidereal time of observation. seen by reference to fig. 13. This is Here C is the center of the earth, and x, y, and z, axes are set up as shown.with 0 as origin, paralell to the great x y and 2 axes whose origin is at the sun. Then x points towards the vernal equinox and z towards the pole of the ecliptic. How let P be a point on the earth*s surface and r its radius from the earth's center. Describe a sphere the about C with radius r. ?] n .x y plane will trace off a A circle x, y, -x, -y, on the sphere, and the plane of the equator another, making an angle € north pole of the earth. and the arc ^3 arc /mix ^S with it. let p be the Pass the meridian ffcPm through P, will be the geocentric latitude of P. The will be the sidereal time of observation, at the place of observationsxpressed in arc, since sidereal time is defined as the Jiour angle of the vernal equinox. Then in the rt. spherical triangle 55 To determine the quadrants of £ and <lr we only need notice that "both of them are always "between 0? and 180*^ so that the cosine of one and the tangent of the other fully fix their values. nhe means are now all in hand for finding ^3/ and thence ^ , that is for locating the actual posi tions of the comet in space at the times of the 1st and last observations, and we can proceed to their actual detefmination. Before doing this, however, it must be decided whether it is worth while to bother with the quantities ^ < i'Xfe ) Tince, as has been stated, the lati tude of the sun is never more than l."0, than a very small quantity, is never more moreover, the earth is but a speck ooiupureu. wi-cn une maginitudes with which we are dealing, and so an observer on its surface is almost at the same position in space as though he were at its center. and zp _'or rough calculation, then, x^ be entirely neglected, and then can y*. > X = o- '> X= * < , so that considerable work is saved. It may be said, too, that even it we do compute and use the values of ^ jo ***** / we will still have only an approximate orbit, because in deriving the formula for the value or IJ, all these were, and had to - be, taken equal to zero. 5 6 - This oannot be denied, ana though the effect in any particular case cannot be pre dicted, it is probable that in many instances the entire omission of the quantities in question will have no sub stantial influence on the final results, but it is equally certain that in the average case a closer approximation will be obtained by taking them into accoxmt than by throwing them out. In this problem, therefore, I will use them, leaving the reader to his own inclinations as to whether he will do likewise in his own work. But with log R the hourly differences change so rapidly that the use of them alone would five slightly erroneous results. tion formula. We must therefore use a n interpola Of these there are many, and a derivation of them and discussion of their relative merits constitutes a large subject in itself. I will therefore use the one given in Campbell's Practical Astronomy, P. 24b. - 58 - If we had. interpolated, hy the hourly differences we shoula have had .0069042. In most oases the hourly differences will give sufficient accuracy, hut the ahove formulae will he used, here for greater certainty. V/e find hy its means in exactly the same manner as for !R,,log R, = .0064539. 59 :no find for the two observations, we must find for each place its. distance from the center of the earth, r, its geocentric latitude, and the sidereal time of observation. The former two are taken from pp. 520 to 524 of the nautical almanac. ^he r's are given direct, while the geocentric latitudes ^ must be found by adding a given correction to the geographical latitudes. Latitudes to the nearest minute will be ample. These, r's are given with the equatorial radius of the earth as a unit, and to reduce them to parts of the radius of the earth's orbit, they must be multiplied s& j fT * 4 y 9 0 . ?) j (, L J *r / -. C i f / oJ» - - I / 0 This .gives for our new log r's G.Ct-f'f’ L-/* i C l f *r2L putting dots over the o's to indicate that 10 is to be substracted from the logs. 7e must next find 3, the local sidereal time at the time of observation. 7/e can do this with sufficient accuracy for these purposes by merely add ing to the hours, minutes and seconds of mean solar time at the place of observation the sidereal time at the be- -60- ginning of the day, which is given on P.Ill in the column "Sidereal time, or Eight Ascension of mean sun." The error made "by this method will never exceed 4 or o minutes, and is here of no consequence. 61 62 63 64 66 6 7 68 6 9 70 71. -72 73 7 4 Fig. 14. 75 F i g . 15. 76 77 The next thing to do is to find J 2 and a *,whioh ) determine the plane or the orbit, and first of all -C2 * The equation of a plane passing through the origin is given in cartesian coordinates by If this plane goes through the points */ y, J , and >r> J y ^ y then If we solve these equations for a and b we get ITow it is the position of the line on which this plane cuts the xy plane which we wish to find, so we set ^ - 0 and get the equation of that line as o . +■ Jfy ~ The tangent of the angle which it makes with the x axis is % = - ^ - C l . values of a and b, we have Substituting the 78 In order to decide which one of these values to give _ Q , we have .only to notioe that the comet is going in towards the sun, since r 3 was less than r; , and that it was when discovered in the fourth quadrant with respect to longitude, and that it was above the plane of the ecliptic. So as it goes arouna the sun it must pass into the second quadrant, and there it will descend below the plane of the ecliptic. The quadrant in which it was when discovered is therefore the one in which to look xor the asoer.ainr node, so we say 79 -80 - Having now found, the plane in which the orbit lies, we must next find the orbit; that is we must obtain the values of CO ana q. open. /or this no direct way seems It proves to be the case, however, that rormulae (c O - V}J can be derived for finding the angles course, v and ' 1where, of v5 are the anomolies of the comet at the two times of observation, and from these far, - &~3) is quickly obtained. /hen another formula gives v; or v in terms of £v t — v^) and other known quantities, and the rest is easy. I shall first show how to find Co-flr and w - . Referring again to figure Id, if S I is the perihelion line, thenw has the place indicated, and draw ing the radius vector Xj S of the comet, ^ . The question now is, whether we should c a l l - i n this position + or -, for the signs of some of the functions in which it appeared would be altered if its sign were changed. Since the majority of comets have their orbits computed before 81 they reach'. ... perihelion, we shall more often avoid bothering with negative angles' if during the computation of an orbit we say that before perihelion v io + , s m after perihelion, minus. This isS,of course, contrary to the most natural way of taking the signs, but the con venience here will justify a departure from the common us'a g e . If v; is to be called + , then, it is obvious that angle and here again we can use our previously found value of S B.to good purpose. It is easy to see that by substitu tion of z for z. , S 3 for S 3. , etc. we have The quadrants of(w>-07/ana can be easily decided upon from the relative positions of y, z/ , >: y, z% , and the line S ± 2 . because since we are finding these angles by their tangents, there is a full 180* between the two possible choices, and one could not easily be mistaken by that much, vive• (o'- 2^ (Or, nd (cu- , their difference gives and the first step is taken. We must next derive the formula for finding v, *■ O - Since a parabola A. - — rlr <Urc> is to be so located that it passes through the points x fy /z/ , and 7: y9 a3 , we must have 82 These are only two equations containing three unknown quantities, q ,v; .and v5 , hut a third equation for determining them comes from the fact that since we know r tf - we Should "be able to eliminate q "by direct divi sion and substitute for vs in terms of ^ and v - v giving an equation in v( and known quantities only. Division and extraction of the square root gives , 83 AS to the quadrant of *L,we know that v, is reckoned either as "betweenC* and 180 always "between -90 will tell us which. , or 0 and -13 u and +90 . -1 is , and the sign ox its tangent -84 - It is apparent from this that #7 is not de termined with exactness, "but the fault is in the data, not in the method. The short arc of a parabola between the 1st and 3 d observations was not great enough to closely fix the perihelion line on purely geometrical considerations. It is possible, however, to introduce another condition, not geometrical, the condition that the times of perihelion computed from v, and v. must be the same, and use this to build up a formula for correction of this preliminary value of v. . But space will not per mit the introduction of that subject in this paper. 85 There is a difference of 18 between these two values, but this is so small as to warrant the belief that it is due to the unavoidable uncertainty about v# which I have spoken of above. So I will take log q = 9.673674 q = .471709 The only remaining element to be found is T, the time of perihelion. This is obtained by determining the time it would, take the comet to go from each of the points where it was observed to its perihelion position, according to the laws of parabolic motion, and adding these figures to the times when it was observed. On p. where 0^1 -■ ? 20 we had -h } and is to be found from the value of v in a Barker's table. Tran sposing the first equation, we have _£i— .— Jjr. f C 0 I I ? 7j 9% - f 2-- r^3 ? b7z3>7 0 / . U This was made upon the supposition that v/ was 4- after pe^helion, instead of before, as we have been using it, but the proper sign for t - T is easily picked out without 86 bothering with the signs for v» for we already know that the comet when observed was not yet at perihelion, and so t(- ? should, be -, and T - t„ +. follows 7e can now calculate as 87 - 8 8 - T'bell’s elements, taken from Astronomishe :Tachrichten 4olO, were merely provisional figures computed on observations taken July 21, 22 and 2 5. Thus the intervals between observations were only about one day, and the great disagreement of his figures with the others is not at all surprising. Young’s elements, from Lick Observatory Bulletin 202, were derived from observations on July 21, Aug. 4, and Aug. 18, so his intervals were 14 days. rillosevich’s elements, from Astronomiche Ilaohriohben 4b36, were c o m p u t e d from a combination of fotir observations, extending over the interval from July 25 to Oct. 5, and are undoubtedly the best of all four sets. It is to be noticed that Young’s elements with intervals of 14 days closely agree with ^lillosevich.,s ; Bbell's, with intervals of 1 day, are widely at variance, while those derived in this paper, with intervals of 5 days, are in fairly close agreement. It appears then, that in this particular case better results would have been obtained if a longer interval had been chosen. The proper length to select, however, depends • upon such a variety of conditions that no rule can be laid down. On the one hand, there are the unavoidable accidental errors of the observations to be considered, and on the other the degree of approximation of that ,0, value of M given by our formulae to the true ratio of /-— . /*' As to errors of observation, it is obvious that the longer the arc swept over by the comet between observations, the smaller the influence of these errors, - 8 9 and with regard only to these, the ideal way to compute an orbit would he to choose one observation when the comet was first discovered, one about midway of its period of visibility, and one just before it went out of sight. The method used here, and any other direct method of calculating an orbit, would.howeve^ he wholly inapplicable to such a case, so a much shorter interval must be taken. In respect to the value of LI at least two opposing factors influence its exactness. The first is the geometrical inaccuracy of the assumption made on p ,42- that areas are proportional to the triangle areas. the sector This assumption will be the more nearly correct as the arc is shorter. On the other hand, the statement that "the areas swept over by the radii vectors of bodies moving about the sun are in proportion to the times" is, as said on p. 2 . 6 , only true if we neglect the attraction of all other bodies but the sun. These attractions are not entirely negligible and the result is that the planets pull a comet slightly away from the true parabolic path assumed for it, and the jj.Lane os and the moon have the same efiect on the .otherwise elliptical orbit of the earth. Besides this, there is the fact heretofore mentioned, that T is computed on the assumption that the comet was seen from the center of the earth, not from its surface, and the apparent place would be slightly different in the two cases. Some of these effects would dictate the choice of long intervals between observations, some of short, and the combined effect is so complex that nothing but.the average 90 or experiences in a great number of cases would indicate what intervals to select. There is one remaining source of error (alluded to on p . 3 0 ) in the elements computed, the so called .planetary aberration, which comes from the fact that the comet was on the observed line of sight not at the instant of observation but at the instant when the light started from the comet to us. Fig. 16. Thus in figure 16, suppose c, and c t to be two successive positions of a comet in space. At a certain tine t . let the comet send out a ray c, E, At this instant the earth may be at some point E . , but light arrives at when the ray of , say. the earth is there to receive it, and to an observer the comet would seem at that instant to be at 3, , when as a matter of tact it had moved on to c r , The error arising from this fact can be entirely corrected by sub. tracting from each observed time, the inter val needed for light to come to us from the comet, thus obtaining the actual times when the comet was on the lines of sight, and using these new times for t. t^ and t5 , in place of the times of observation. But inasmuch as the 91 distance of the comet is not. known- until an orbit has heen computed, we must first calculate an orbit, neglecting the planetary aberration, thence obtain and > the.distances of the comet from the earth, from these estimate the corrections to the three times, apply them, ana recompute the orbit. 'i’he second computation is far less difficult than the first, for one can use the first as a basis and simply make small changes in the last figures of the logs* without bothering to look.the logs.up all over again. A recomputation of the present orbit in this way turnishes the following elements, which are given in parallcll columns with*, original for the sake of comparison. It appears, then, that the difference is small, but for an orbit which shall represent positions of a comet with the greatest possible accuracy it is necessary to take it into account. To find how nearly the above final orbit re presented actual positions of the comct, the position of the latter was computed for the -morning of ITov. 1, 1911. three months and ten days after discovery. The methods by which this was aone cannot be explained here, so only the 92recults will be given, as follows It is here seen that the error in oc is nearly 1(T, or 2' *, and that in S over 2/3 of a degree, so it seems that while an orbit computed on observations at such an early aate after the discovery of a comet as was this one will represent its general positions through the rest of its appearance, and show what sort of a display it will make, the orbit will not give accurate places throughout as long an interval as three months. Bibliography. The work chiefly consulted. in the preparation of this paper s bee n the "She ore cishc Astronorie* ox edition of I8‘ JC J, rna That aouii is o Dr dtiongl '.linkerfues, reo/o- cnaea to whoever aerires ro go into This sub.i? oa ■ ■ rreater length. The explanations re given in as simple ana thorough c form ce po siole, ana the desire to build everything up xirrol" and clesrl; irom the fundamentals is not xoregone :or rhr s ko of a little greater deftness in rh*- demonstrations. ’.Tatson's mheoroticrl Astronomy js i w o r i n English on the sane subject, but. because o- iTs age ; s' well as other reasons I cannot rco/zror.rcna it.. - One desiring t 9 3 - A 'iv .m a y or orbit computation, taking up methods that have been, !c we&las Those th.- t re at present ir, uc. , n;.y consul: the t'ollo' ing works: "De Determination aec Orbits clec 3ometes et Planetes.1' by ''ppolser. i- 'theoria "otus Corporum Doelestiun,'T by Gauss, an ’"nglish t:. anslrti or. oi which was oubli she a by I H tlr , Brown, Tr, Jo. , or Dost on, in 185 7 . Laplace’s kecaniqué Celeste, 3ook T r. In -Tewton1s T-rincipia there is given graphic r.cthoa or linding the orbit or a cornet, which, so far t] c I know, is ïirst successful sol ition or the -problem. Te applies it to several 01 the oomets ox his-time., ana v it;■ rw-oa results. In Popular Astronomy of 1909 there was published, ser ially ■ modern graphic • : 4hoa, aevisea by 'n . 0 . Penrose, Which will furnish an interesting exercise, ana will sometimes q s* Its value beginner an exact vis u 1 :an: or rl is cl ' v dporti< ns or his erbiJ ; r.sid- trom tk.t it has little worth except es a curiosity/.