ASTRONOMICAL NOTES

Evening Star, Issue 23128, 30 November 1938, Page 18

ASTRONOMICAL NOTES

THE SKIES IN DECEMBER . (Written by A. 0. C. Crust, M.Sc., lor the ‘ Evening Siar.’l POSITIONS OF THE STARS. Local Sidereal Time 4h, Latitude -ICdog S. The following .positions of the stars hold good locally four minutes earlier each evening, as at midnight, December 2-3, n.(M) p.m. on December 17, and 10.UO p.m. on January 1, 11130, in New Zealand summer time:— Achernar is now S, W., CCdeg, Fomalhant W. by S., 30deg, Aldebaran X. by 27dcg., Rigel N.N.E., oOdeg, Botelgense N.E. by N., 3’2deg., Pollux N.E., Ideg, Procyon N.E. by E., 20deg, arid Sirius in the same direction, at an altitude of 46deg. Canopus, in tbe S.K. by 10., sha-es with Achernar the distinction of being the highest of our bright stars, for his altitude is also 66dog Alpha Crueis, the brightest star in the Southern Cross, is now S.S.E., 27deg, and thus at the same altitude as Aldebaran, while Alpha Ccntauri is S. by 11..,I 1 .., ISdeg, thus practically level with Procyon. The Moon at midnight, December 2-3, will he situated in the X.W. by W., 19deg above the horizon. She will pass Gdcg north of the iplanet Saturn on the afternoon of December 3, and 3.sdcg N. of Aldebaran on that of the 7th, while Full Moon occurs the same evening, the Moon’s position at 11.40 p.m. being N. by JO., 22deg. The waning moon will pass lldeg S. of Pollux on the afternoon of December 10, last quarter occurs on the afternoon of the 14th, and on that of the 17th onr satellite will pass 2dog S. of the planet Mars. On the mornings of the 19th and 21st she will pass the planets Venus and Mercury respectively, being only half a degree away when passing the latter planet. New Moon falls on the forenoon of December 22. onr satellite will pass Cdeg N. of the planet Jupiter at noon on the 27th, and her position at 10.20 p.m.'that day will be W. lOdeg. She will pass 28dog X. of Fomalhant at noon on the 28th, first quarter will occur on the forenoon of December 30, and on the same evening the Moon will again be placed Gdeg X. of the planet Saturn. Summer begins at midnight December 23-24, when the solstice occurs. The planet Mercury will be stationary in the evening sky on December 4, in inferior coni unction with the Sun on the loth, and stationary in the morning sky on December 24. The planet Venus is now a most brilliant object in the early evening sky. She will be stationary on December _ 9 and will attain to her greatest brilliancy on December 26. In quite small telescopes she may now be seen in her crescent phases. The planet Mars la in the morning sky during the month, and may be seen by very early risers. The planet Jupiter is declining rapidly from the favourable position he has occupied during recent months. Ids position at 4h S.T. during December being W. by S., 11 to 13deg. The planet Saturn is still moderately well placed for observation in the evening sky. Ids position during the month at 4h S.T. being X.W. by W., 26deg. He will bo stationary in the evening sky on December 15. Saturn’s prominent satellite. Titan, will be at greatest elongation oast on the afternoons of December 3 and 19, and at greatest elongation west on December 11 and 27.

From the Queen’s Gardens, Dunedin, at 4h S.T., Fomalhaut will appear moderately low and Jupiter low in the direction of Customhouse Square, and Aldebaran rather low in the direct'jn opposite to Vogel street. At 11.40 p.m. on December 7, after her conjunction with Aldebaran, the Moon will also be in this position, Rigol is rather high over the Law Courts, ißetelgeuse moderately low over Anzac Square, Procyon rather low and Sirius at a moderate altitude over Lower High street, and Alpha Crucis rather low in the direction opposite the Burns Hall.

THE POSITION OF A BODY IN AN ELLIPTICAL ORBIT.

In last month’s notes a general description of elliptical orbits was given, but the method of finding the position of a star in its orbit was not explained. Kepler’s second law states that the radius vector, which in the case of a planet is the line joining that planet to the sun, sweeps ever equal areas of the planet’s orbit in equal times. It follows, therefore, that when the radius vector is shortest—i.e., at periastrou for a star or perihelion for a planet, that body’s orbital velocity is at a maximum, while at apastron (or aphelion), the orbital velocity is at a minimum. Though this principle sounds very simple, it is by no means easy to determine the position of a body in its orbit at any given time by calculation. The problem is actually to find the length of the radius vector and the angle at which it is inclined to the major axis of the orbit at any given time. This angle of inclination of the radius vector to the major axis is called the “ true anomaly.” Fortunately, both the radius vector and the true anomaly are related in a comparatively simple manner to the “ eccentric anomaly,” which we may denote by the symbol “ U,” and from which values of time, T, measured from periastron. are given by the equation, U minus C sin U equals NT. In this equation N is the mean daily motion in degrees, while C is ithc ellipticity in degrees, obtained by multiplying the eccentricity E by the number of degrees in one radian, which is approximately 57.296 deg. Now. as it is essential for purposes of observation to know a series of positions of the revolving body, it is quite convenient to commence by taking a series of values of U and computing the corresponding values of T. As an illustration, let us take a bodv moving in an orbit of eccentricity 0.50 in a period of ISO days, at a mean distance of 1.00 astronomical units from its primary. For this orbit the value of C is one-half of 57,290. or 2S.64.Sdeg. while N is 060 divided hv 180. or 2.0. Our equation thus hec'-mes U minus 25.G18 sin U equals 2T. To illustrate the inequality of the motion of the bodv. let us solve this equation for the case when IT equals OOdeg. Sine OOdcg equals 1.00. so here 2T equals 00 minus 28.C15. hence T equals 30.G7G davs. As it always takes half the period, in (his case 90 days, to go from periastron to apastron, the position furthest removed from the primary. and since IT is OOdeg at the halfwav position, on the minor axis of the ellipse, we see that the second half of the iouruev tabes nearly twice as long as the first half. The equation '■nnueet'ug U with the radius vector. 11, is the following;— 11 equals A minus AF cos IT. If we take the values given for mir exnirmle. this becomes: I? equals T 00 minus 0.50 eos U. At nci-iastron F is zero and its cosine is 1.00. hence 11 0.50. while at aoastron F equals ISOdeg. its eo-i'ie is m : "us 1 Oft. hence R becomes 1.50. For F OOdeg, cos U is zero, ami 11 is 1.00.

The equation for the true anomaly would be difficult to print in its original form, so that here we shall introduce three new symbols: Let U be the eccentric anomaly as before, and Y, one-half of this angle, while V is the true anomaly, ami W one-half of the true anomaly. Wo require a constant F, derived from E as follows:—Divide one plus E by one minus E, and obtain the square root of this ratio. In onr example, F is the square root of (1.5 divided by 0.5) —i.e., the square root of 3, or approximately 1.732. The equation may now be written, tan W equals F tan Y. Using our illustration, tills becomes, tan W equals 1.732 tan Y. At periastron, both angles are zero, and at apastron both arc ISOdeg. A practical difficulty is apt to emerge when accurate values are desired near apastron, as in the case of Alpha Pavonis, whose principal eclipse terminates near that epoch, for the tangents of Y and W become very large and change very rapidly with small changes in the angles. When U is OOdeg. Y is 45deg, so that tan Y is 1.00. Then in onr example, tan W is 1.732, so that W is GOdeg, and therefore V is 120 deg.

To obtain a general idea of the motion in an elliptical orbit, we need only work out those equations for every 30deg increase in the value of U. Many problems of this type can ho solved with considerable accuracy by the use of a slide-rule and graphs constructed for the relationships of U and T, while any degree of accuracy likely to he required may he obtained by the use of suitable logarithmic and trigonometrical tables. THE NEW ECLIPSING VARIABLES IN DECEMBER. The reflection effect of Lambda Eridani ends 3.4 minutes earlier each evening—c.g., December 0:23:47, 16:23:11, and 26:22:30. The total eclipses of the companion of Lambda Eridani occur in the early mornings—e.g., December 7:03:26, 17:02:50, and 27:02:15. The companion of Tan Orionis is eclipsed 15.1 minutes earlier each day, so that, though the eclipse of December 5 begins at 2:02 a.m., that of December 14 begins at 23:31 (11.31 p.m.), and that of the 24th at 21:00, which is about as early as it is possible to observe in midsummer in onr latitudes. The terminations of these three eclipses are predicted for December 3:04:21, 15:01:49, and 24:23:18 respectively. An eclipse of Mu Ononis will begin on December 5:21:36, while twn_ others will terminate on Decebmer 15:04:05 and 24:01:44. Towards the end of the month three eclipses of 32 Orionis will bo visible, commencing on December 18:21:35, 23:23:48, and 29:02:02.

The observation of colour variations is good sport, for one never knows when a hitherto well-behaved comparison star will go “ off-colour ” and so betrav itself as an eclipsing variable, which has eluded discovery bv the methods employed overseas. It is not exactly a sedentary form of sport, either, for observations should be taken at a rate of about one every two minutes, tin’s generally involving fairly active movements on the part of the observer, which compare not unfavourably with those of tennis or swimming. As the year draws to a close the Star Colour Section finds itself invading several realms of astronomical research, such as that of stellar orbits, and promising valuable assistance in _ every problem to which its attention is turned. With its steadily mounting record of discoveries, its observers should feel the stimulus of success, perhaps, far more than the members of nnv other astronomical group.

The incompleteness of our knowledge of several eclipsing variables in the Orion region has been stressed already. Attention should bo given to Psi, 6G and 68 Eridani. 37 and Upsilon Orionis. Secondary minima for the remaining variables in this region have yet to be detected.