A Correction of the Theory of the Motion of the Satelite. of Saturn, by That Ingenious Astronomer Mr. Edmund Hally

A Correction of the Theory of the Moti- on of the Satellite of Saturn, by that In- genious Astronomer Mr. Edmund Hally. Sir, I here send you an Astronomical account of the most remote of all the Planets of our Vortex, and withal of the most inconsiderable, I mean of the Satellite of Saturn, discovered in the year 1655 by Monsieur Christian Hugens of Zulichem, who in that accurate Treat- ise of his Systema Saturnium, from pag. 25 to 34 gives us the Theory of its Motion as well as the shortness of the interval of time between his Observations would admit; and since him I know none that have gone about to improve the said Theory. The late Conjunctions of Jupiter and Saturn giving me frequent occasions of viewing them both, with a Telescope that I have of about 24 foot, and pretty good of that length, I easily remarked this Sa- tellite of Saturn, and having found it, in a convenient position to de- termine its place, I perceived that Hugens's numbers were consi- derably run out, and about 15 degrees in twenty years too swift; this made me resolve more nicely to enquire into its period; and ac- cordingly I waited till I had gotten a competent number of Observa- tions, the most considerable whereof are these. 1682. November 13° 13' 00" PM. the Satellite appeared on the North side of Saturn, and a perpendicular let fall from it on the trans- versal diameter of the Ring, fell upon the middle of the dark space of the following Ansa; and the same night 19° 00' it had past the Conjunction, and the perpendicular fell exactly on the Western edg of the Globe of Saturn, as in Tab. 2. Figure 8. The Northern Lat- itude, and retrograde motion, made it evident that the Satellite was then in Perigee. Again Again November 21°. 16h. 15'. this Satellite of Saturn was on his South side, the perpendicular on the line of the Anse fell on the middle of the dark space of the Western Anse, and the same night 19h 00' the perpendicular fell precisely on the Center of Saturn, and the distance therefrom was somewhat less than one diameter of the Ring (as in Fig. 9.) by this it was evident that the Satellite was in Apogeo. I observed it in Apogeo again on the 24th of January 1683, at 8h. 00' PM. the perpendicular on the line of the Anse fell exactly on the Western limb of the Globe of Saturn, and at 9h 30' PM. the said perpendicular fell within the Globe more than half way to the Center, and the distance from the line of the Anse towards the South seemed much about one diameter of the Ring, Fig. 10. Lastly, February 9°. 1683. 8h 10' PM. it was again in Apogeo, and I could by no means discern towards which side it inclined most, nor whether the transvers diameter of the Ring, or the distance of the Satellite therefrom were the greater; so that at that time it was precisely Apogeon. Fig. 11. To compare with these, I chose two out of those of Monsieur Huygens, which seemed the most to be confided in; the first made 1659, March 14°. 12h 00'. at the Hague; when the Satellite appeared about one diameter of the Ring under Saturn, but it was gone so far to the Westward, that he concluded, that about four hours before, or 7h. 40'. at London, it had been in Perigaeo. Fig. 12. Again March 22°. 1659. 10h 45'. the Satellite was a whole diameter above the line of the Anse, and the perpendicular thereon fell nearly upon the extremity of the Eastern Anse. See Fig. 6. I could wish that we had some intermediate Observations, but there are none extant that I can hear of; so I proceed to the search of this Satellite's period. By the first of my Observations it appears that the Satellite was in Perigaeo 1682 November 13°. 17h 00', circiter, at which time Saturn was 3°. 21°. 39' from the first Star of Aries, in the Ecliptick, but the Earth reduced to Saturn's Equinoctial, and the Satellite was 9°. 23°. 46'. a 1° * ν. And March 4° 1659. 7h. 40'. Saturn's place in the Ecliptick was 6°. 0°. 41', but the Earth reduced and consequently consequently the Satellite in $11^\circ 28' 18''$ à prima Stella Arietis. The interval of time is 8655 daies, 9 hours, 20 minutes; in which the Satellite had made a certain number of Revolutions to the fixt Stars, and besides $9^\circ 25' 28''$, or 295 degrees, 28 min. whose Complement to a Circle $64^\circ 32'$. So that 8658 dates, $5^\circ 56'$ or 12467876 minutes of time, is the time of some number of entire Revolutions; and dividing that interval by 15 daies 22 hours, 39 minutes, or 22959 min. (the Period of Hugens) the Quotient 543 shewes the number of Revolutions; and again dividing 12467876 min. by 543, the 22961 $\frac{1}{2}$ min. or 15 daies, 22 h. 41' 6'' appears to be the true time of this Satellite's Period. Hence the diurnal motion will be $22^\circ 34' 38'' 18''$, and the Annual besides 22 Revolutions $10^\circ 20' 43'$. Having made Tables to this Period, I found that in the Apogon Observation of Hugens the Satellite was above 3 degrees faster than by my calculus, and that in the three other Observations of my own being likewise in the superior part it was about $2\frac{1}{2}$ degrees slower than by the same Calculation. Now 'tis evident that these differences must arise from some Eccentricity in the Orbite of this Satellite, and that in March 1659, the Apocronion (as I may call it,) was somewhere in the Oriental Semicircle, and that in November 1682 it was in the Western Semicircle, and supposing the Apocronion fixt, it must necessarily be between $9^\circ 23' 46''$ and $11^\circ 28' 18''$. a $1^\circ \ast \gamma$ that being the common part between those two Semicircles: and because the difference was greater in Hugen's Observation than in Mine, 'twill follow that the Linea Apsidium, or Apocronion, should be nearer to $9^\circ 23' 46''$ than to $11^\circ 28' 18''$. I will suppose $10^\circ 22' 00''$ à prima Stella Arietis, (which happens to be also the place of Saturn Equinox,) and the greatest equation about $2\frac{1}{2}$ degrees. Upon the score of this inequality the mean motion of the Satellite will be found about $2^\circ 45'$ slower in $2\frac{1}{2}$ years, or 7 min. in a year, whence I state the Annual motion $10^\circ 20' 36''$ above 22 Revolutions, and the correct Epoch for the last day of December 1682 at Noon in the Meridian of London $9^\circ 10' 15''$. a $1^\circ \ast \gamma$, from which Elements I compose the following Table. Tabula Motus Medii Satellitis Saturnii, ab Hugenio inventi, a prima * r. | Anno | Epocha | Mot. Med. | Mot. Med. | Mot. Med. | Mot. Med. | |------|--------|-----------|-----------|-----------|-----------| | 1641 | 8. 29. 17 | 10. 20. 36 | 10. 22. 35 | 10. 56. 31 | 29. 10 | | 1661 | 10. 14. 10 | 9. 11. 12 | 1. 15. 5 | 1. 53. 32 | 30. 6 | | 1681 | 11. 29. 3 | 8. 1. 48 | 2. 7. 44 | 2. 45. 33 | 31. 7 | | 1682 | 10. 19. 39 | 7. 14. 59 | 3. 10. 18 | 3. 46. 34 | 31. 59 | | 1683 | 9. 10. 15 | 6. 5. 35 | 3. 22. 53 | 4. 42. 35 | 32. 55 | | 1684 | 8. 00. 51 | 4. 26. 11 | 4. 15. 28 | 5. 39. 36 | 33. 52 | | 1685 | 7. 14. 2 | 3. 16. 47 | 5. 8. 2 | 6. 35. 37 | 34. 48 | | 1686 | 6. 12. 2 | 2. 26. 57 | 6. 0. 37 | 7. 32. 38 | 35. 45 | | 1687 | 5. 12. 2 | 1. 20. 23 | 5. 23. 12 | 8. 28. 39 | 36. 41 | | 1688 | 4. 12. 2 | 10. 11. 9 | 7. 15. 46 | 9. 24. 40 | 37. 38 | | 1689 | 3. 12. 2 | 11. 1. 45 | 8. 21 | 10. 21. 41 | 38. 34 | | 1690 | 2. 12. 2 | 10. 14. 56 | 9. 0. 55 | 11. 17. 42 | 39. 31 | | 1691 | 1. 12. 2 | 9. 5. 32 | 9. 23. 36 | 12. 14. 43 | 40. 27 | | 1692 | 10. 11. 9 | 8. 26. 8 | 10. 16. 5 | 13. 1. 44 | 41. 24 | | 1693 | 9. 11. 9 | 7. 26. 8 | 11. 8. 39 | 14. 7. 45 | 42. 20 | | 1694 | 8. 11. 9 | 6. 16. 44 | 12. 1. 46 | 15. 3. 46 | 43. 17 | | 1695 | 7. 11. 9 | 5. 29. 54 | 13. 23. 48 | 16. 9. 47 | 44. 13 | | 1696 | 6. 11. 9 | 4. 20. 3C | 14. 16. 23 | 17. 5. 48 | 45. 10 | | 1697 | 5. 11. 9 | 3. 16. 19 | 15. 8. 58 | 18. 1. 49 | 46. 6 | | 1698 | 4. 11. 9 | 2. 26. 12 | 16. 1. 42 | 19. 1. 50 | 47. 3 | | 1699 | 3. 11. 9 | 1. 13. 31 | 17. 1. 43 | 20. 1. 51 | 48. 56 | | 1700 | 2. 11. 9 | 0. 23. 24 | 18. 1. 44 | 21. 1. 52 | 49. 52 | | 1701 | 1. 11. 9 | 11. 10. 43 | 19. 1. 45 | 22. 1. 53 | 50. 49 | In Anno Bifextili post Februarium adde unum diem motum ei competentem. There suppose the Linea Apsidum fixed, as having no arguments from Observation to prove the contrary, tho it be very probable that as the Apogeeon of our Moon has a motion about the Earth in about 9 years, so that if this Satellite ought to have about Saturn, but with a much longer Period, which future Observation may discover. The distance of this Satellite from the Center of Saturn seems to be much about 4 Diameters of the Ring, or 9 of the Globe, and the plane wherein it moves very little or nothing differing from that of the Ring, that is to say, intersecting the Orb of Saturn $4^\circ 22'$ and $10^\circ 22'$. * $\gamma$, with an Angle of $23\frac{1}{2}$ degrees, so as to be nearly Parallel to the Earth's Equator; whence the Latitude of the Apogeeon Semicircle from $4^\circ 22'$ to $10^\circ 22'$ of Saturn's Longitude, will be Northern, and of the other Semicircles Southern; and the contrary in the other half of Saturn's Longitude, to wit, from $1^\circ 22'$ to $4^\circ 22'$ of his distance from the first Star of $\gamma$. It follows now to show how by the help of this Table to compute the place of this Satellite, to any time required. First we must have the true Longitude of Saturn from the Earth, and numbered from the first Star of $\gamma$, (or rather the place of the Earth viewed from Saturn together with its Latitude from the Orb of Saturn, but that being never fully $\frac{1}{2}$ of a degree we neglect it as a nicety) and therefrom subtract $10^\circ 22'$. there remains the distance of Saturn from this Equinoctial point, with which distance as with the Longitude of the Sun, take out the Right Ascension and Declination thereof ($23\frac{1}{2}$ degrees being the obliquity common to both) and to the Right Ascension adding $10^\circ 22'$. the sum shall be the Longitude of the Satellite's Apogeeon. Then say, As Radius to sine of the Declination, so 8 to the greatest Latitude in Apogee, or Perigee in the parts of the semidiameter of the Ring. Next collect the middle motion of the Satellite, and from it subtract $10^\circ 22'$; the remainder shall be the mean Anomaly, with which in the Table of the Moon's primary Equation; take out the Equation answering thereto, and the half thereof added or subtracted to or from the middle motion, according to the Title, gives the true motion of the Satellite, from which subtract the Apogeeon, and if the remainder be more than 6 Signs, the Satellite is Occidental; if less Oriental, and as Radius to Sine of the remainder, so 8 to the Semidiameters of the Ring; or 18, to the Semidiameters of the Globe; that the Satellite is to the Eastward or Westward of the center of Saturn, according to the afore-going Precept. Lastly, As Radius, to Co-sine of the said remainder, so is the greatest Latitude from the line of the Anse, to the Latitude sought. Here Note, that I purposely neglect the inequality of the distance, arising from the Eccentricity, as being too small to be any way observable. Lastly to clear all difficulties that may arise to them that are but little versed in this sort of Calculation, I have added Two Examples of the work, that where the Precept may seem obscure it may be thereby illustrated. Anno 1657 Maii 19 st. n. Hugens Observed the Satellite very near to Saturn on the Western side, and very little above the line of the Ansa. I suppose this about 10 h. p. m. Anno 1658 Martii 11°. 10 h. st. n. he Observed it again, and said of it, difficilè conspiciebatur, quippe propinquius admodum Saturno, Orientem spectabat, eratq; Anserum linea aliquanto inferior & quasi sub Saturno transiturus. Let us now Calculate to these two times. | Year | Month | Day | Hour | Minute | Second | |------|-------|-----|------|--------|--------| | 1657 | Maii | 9° | 9 h | 40' | Londini| Saturni Locus | Hour | Minute | Second | |------|--------|--------| | 28 | 57 | h a 1° * γ | Hour | Minute | Second | |------|--------|--------| | 5° | 0 | 32 | Equinot. sub. | Hour | Minute | Second | |------|--------|--------| | 10° | 22 | 00 | hab Equinot. | Hour | Minute | Second | |------|--------|--------| | 6° | 32 | Ascen. Recta. | Hour | Minute | Second | |------|--------|--------| | 6° | 7 | 50 | Apogaeon. | Hour | Minute | Second | |------|--------|--------| | 4° | 29 | 50 | Declin. Aust. | Hour | Minute | Second | |------|--------|--------| | 3° | 23 | Med. Mot. Satel. | Year | Month | Day | Hour | Minute | Second | |------|-------|-----|------|--------|--------| | 1641.| Maii | 6° | 9° | 14 | | | | | 6° | 23 | 12 | | | | | 9 h | 40' | | Long. Med. Satel. | Hour | Minute | Second | |------|--------|--------| | 4° | 10 | 42 | Apocron. | Hour | Minute | Second | |------|--------|--------| | 10° | 22 | 00 | Anomalia | Hour | Minute | Second | |------|--------|--------| | 5° | 18 | 42 | Æquatio sub. | Hour | Minute | Second | |------|--------|--------| | 3° | | Long. Ver. Satel. | Hour | Minute | Second | |------|--------|--------| | 4° | 10 | 11 | Apogaeon | Hour | Minute | Second | |------|--------|--------| | 4° | 29 | 50 | Residuum | Hour | Minute | Second | |------|--------|--------| | 11° | 10 | 21 | h. e. ante Apogaeum | Hour | Minute | Second | |------|--------|--------| | 19° | 39 | ergo 2° Semid. Annuli ad occasum | Hour | Minute | Second | |------|--------|--------| | 1° | | & 2° ad Boream. In each agreeing exactly with the Description and Figure of Monsieur Hugens. I here call the Plane of this Satellite's Orb, which hitherto I suppose the same with that of the Ring, Saturn's Equinoctial, not that any discovery hath been able to prove that the Axis of that Globe is at right Angles thereto, but because it has pleased Mr. Hugens to call it so, and likewise because it is so nearly Parallel to our Globes Equinoctial; Nevertheless to speak my Opinion, I believe that the Axis is inclined, and that not a little, to the Plane of the Ring; for as the reflection of the Sun's light from the Ring is a great convenience to that Hemisphere of Saturn, which beholds its illuminate side; so the other Hemisphere is very much incommoded by the shadow of the Ring, which for many Months, and in some Parallels for several Years, occasions a continual Night by the interception of the Sun's beams, which is a consequence that demonstratively follows the position of the Ring in the plane of Saturn's Equator. Now this great inconvenience would be in some measure relieved by the oblique position of the Axis, for then the Parallels of Latitude intersecting the plane of the Ring, many and in most cases all of them, might for some time in every Diurnal revolution of the Globe free themselves from this Eclipse, which otherwise were sufficient to render this Globe of Saturn unfit for any settled habitation; but this is but conjecture. The other Two Satellites of Saturn discovered by Signor Cassini at Paris Anno 1672 and 1673, I must confess I could never yet see. I have been told that they disappear for about $\frac{2}{3}$ of Saturn's revolution, and were only to be seen when the Ansae were very small, it being supposed that the light which proceeds from the Ansae when considerably opened might hide these Satellites. In the year 1685 when the Ansae will be quite vanished, will be a proper time to look after them, that so we may bring their Motion to Rule, and know where to find them, for want of which knowledge 'tis likely they are at present not to be found. Obser.