Mechanics of the Solar System: An Introduction to Mathematical Astronomy

By J.A. Evans

This book develops methods of computing astronomical phenomena from basic ideas. The position of a celestial body is defined by a vector, with components referred to a system of coordinate axes. The relations between various systems in regular use by astronomers are described. In cases where two systems differ in spatial orientation, they are related by a rotation matrix. These matrices are discussed in considerable detail in the mathematical notes.

Other topics discussed include: Kepler's Laws and the dynamics of planetary motion, Precession and Nutation, transits of Venus and Mercury, Lagrange points.

While no previous knowledge of Astronomy is necessary, it is assumed that the reader is familiar with elementary algebra, trigonometry and calculus.

• Language: English
• Publisher: Brown Dog Books
• Release date: Mar 12, 2021
• ISBN: 9781839522352

Book preview

exercises

Introduction

Mathematics and astronomy have been closely linked since antiquity, and mathematics has sometimes signposted major conceptual developments in astronomy. In the 16th century it was Copernicus’ discovery of serious inaccuracies in the predictions of Ptolemy’s algorithms which led him to his radical revision of the structure of the cosmos. In the next century Kepler used Brahe’s data on planetary positions to show that the planets move on elliptical, rather than circular, orbits. Later in the same century Newton demonstrated mathematically that, according to his laws of motion and gravity, a single planet would move in an elliptical orbit around a single star. Newton’s work forged the first direct link between astronomy and physics. Kepler’s empirical laws of planetary motion are all derivable from Newtonian mechanics applied to a single orbiting planet.

This book presents numerous demonstrations of how mathematics can be used to compute planetary phenomena. While no prior knowledge of astronomy is required, a familiarity with basic algebra, trigonometry and calculus is assumed. Any serious work in this subject, such as the computation of planetary ephemerides, demands a familiarity with computer programming in a high level language such as Fortran. The text relies heavily on vector analysis and matrix algebra, and these areas of mathematics are treated in some detail in Chapter 10. For some applications it is useful to treat vectors as matrices with a single row or column, creating a distinction between ‘row’ and ‘column’ vectors, which are written horizontally and vertically respectively and enclosed in square brackets. In other applications this distinction is unimportant and all vectors are written horizontally in round brackets. It is not anticipated that this will lead to any confusion. Equations are numbered independently within each chapter, so that they can be referenced in the text. In cases where it is necessary to refer to equations in other chapters, the chapter number is placed in square brackets together with the equation number; e.g. [8(6.3)] refers to equation (6.3) in Chapter 8.

The main tool for calculating planetary positions and velocities is VSOP, as described in Chapter 4. The VSOP 87 data files can be accessed via the link ftp://ftp.imecce.fr/pub/ephem/planets/vsop87.

In addition to orbital elements, there are five other versions of the theory, designated A–E as indicated below.

A. heliocentric rectangular coordinates ref. to equinox and ecliptic of J2000.

B. heliocentric spherical coordinates ref. to equinox and ecliptic of J2000.

C. heliocentric rectangular coordinates ref. to equinox and ecliptic of date.

D. heliocentic spherical coordinates ref. to equinox and ecliptic of date.

E. barycentric rectangular coordinates ref. to J2000.

The data given in the appendix of Reference 1 is a subset of that in version D.

The site also lists a Fortran source code which uses VSOP87 to calculate planetary positions.

Chapter 1

Time and Space in Astronomy

Astronomy is the science which studies the distribution and movements of stars, planets, comets etc. in the cosmos. Before going on, in later sections, to describe how the locations of celestial objects in space are recorded, we begin by discussing the measurement of time.

1.1 Time

Any regular periodic phenomenon can be used to measure time. As most human activity is carried out in daylight, and the most important in early civilisations, viz. agriculture, depends critically on the seasons, the Sun is a fairly obvious choice of timekeeper. However, many primitive cultures used the Moon and even hybrid combinations of Sun and Moon to measure long periods of time. This sometimes resulted in a chaotic situation with the calendar seriously out of step with the seasons. In a successful calendar, the equinoxes and solstices should occur on or very close to the same dates each year.

Julius Caesar found it necessary to reform the Roman calendar, which, as his biographer Suetonius wrote: ‘the pontifices had allowed to fall into such disorder, by intercalating days or months as it suited them, that the harvest and vintage festivals no longer corresponded with the appropriate seasons’. In order to achieve his reform he added 80 days to 46 BCE making it, with 445 days, the longest year in history. His new Julian calendar was inaugurated on the designated first day of 45 BCE January 1. The Julian calendar assumed that the mean tropical year (i.e. from vernal equinox to vernal equinox) contains exactly 365¼ days, and introduced the familiar leap-years of 366 days so that each four-year period contained 365 × 3 + 366 = 1461 days. The Julian calendar was in general use throughout the Christian world until the late 16th century and until much later in some countries.

1.2 Universal Time

The time used for everyday life in general, and astronomical observations in particular, must depend ultimately on the Sun, i.e. on the Earth’s diurnal rotation. This need is met by Universal Time (UT) which is based on observations of the transits of stars across the local meridians of observatories across the globe. These are converted from sidereal to solar time and corrected for the longitude difference from Greenwich. Such is the accuracy required that this last step is complicated by the fact that the longitude of a place on the Earth’s surface varies periodically by a few tenths of an arc-second. This is due to tiny motions of the Earth’s geographic pole relative to its rotation axis. Various periods are involved but the best known is the ‘Chandler Wobble’ with a period of about 430 days.

The Sun is not a perfectly regular timekeeper because its apparent diurnal motion across the sky not only depends on the Earth’s axial rotation, but is affected by its orbital motion around the Sun. The orbital motion causes significant irregularities and, when these are ironed out, what is left is the mean sun.

Universal Time (UT) is defined by the motion of the mean sun through its Greenwich Hour Angle (GHA). The mean sun transits (i.e. crosses) the prime meridian at Greenwich at 12h.00 UT and one hour later, at 13h.00 UT, it transits the meridian at longitude W15° at which instant its GHA is 15° or 1 hour. Thus UT runs 0 to 24h from midnight to midnight and is equal to GHA (mean sun) plus 12 hours.

The difference GHA (Sun)–GHA (mean sun) multiplied by 4 to convert degrees to temporal minutes, where the first term refers to the real Sun, is called the Equation of Time (EOT) and is a measure of the irregularity of the solar clock. At certain times of the year, the Sun lags or leads the mean sun by more than a quarter of an hour; but the EOT vanishes four times each year. Figure 1.1 is a plot of the EOT on Christmas Day 2010 between 6 am and 6 pm. It can be seen that it vanished at about 11:21 on that day and was decreasing at a rate of 30 seconds per day. The EOT is discussed in more detail in Chapter 9.

Figure 1.1

The equation of time on 2010, December 25.

Currently, a continuous, uniform timescale based on atomic clocks, which use a microwave transition in caesium to establish a frequency standard to an accuracy of 1 part in 10¹³, is available for scientific time-keeping. International Atomic Time (TAI), is established by the Bureau International de l’Heure (BIH) in Paris through data from atomic clocks based in about 40 laboratories in different parts of the globe. Coordinated Universal Time (UTC) is another timescale based on atomic clocks. The rotation of the Earth is very gradually slowing down and therefore cannot be permanently in step with TAI. The process of slowing is irregular and unpredictable, and a ‘leap second’ is occasionally added to the time lag, currently about 34 seconds, between UTC and TAI. This is usually done by giving the last minute of June 30 or December 31 one additional second. The difference between UT and UTC is monitored by the International Earth Rotation and Reference Systems Service (IERS), to keep the difference to less than 0.9 seconds.

It is recognised that the discrepancy between UT and atomic time will become a serious problem in the future, when the length of the day will have increased sufficiently to require the addition of a leap second twice every year. At present there is no agreed solution to this problem.

The time used by astronomers in the calculation of planetary ephemerides is dynamical time (TD) given by TD = TAI + 32s.184. This continues smoothly from ephemeris time (ET) which was used before 1977 and was based on the motion of the Moon. Dynamical time is usually calculated from TD = UT + ΔT, in which ΔT changes in an unpredictable manner from year to year. Its current value (2019) is 69.7 seconds.

1.3 The Solar year

The seasons on the Earth are governed by the apparent oscillatory motion of the Sun with respect to the Equator. This occurs because the ecliptic, i.e. the plane of the Earth’s orbit and the Earth’s equatorial plane are inclined to each other at an angle ϵ = 23° 26'. This implies that the Earth’s polar axis is tilted away from the North Ecliptic Pole (NEP) at the same angle. The angle ϵ, which is called the obliquity of the Earth’s axis, like most of the ‘constants’ in astronomy is subject to gradual change over time. The tilt of the Earth’s axis has the effect that, as the Earth travels round its orbit, the latitude of the point on the Earth’s surface where the Sun is directly overhead, at any given longitude, oscillates between N23° 26' (the Tropic of Cancer), and S23° 26' (the Tropic of Capricorn). The Sun’s arrival at the northern tropic is termed the summer solstice and occurs around June 21 each year. The winter solstice, when the Sun is above the southern tropic occurs around December 21. The vernal equinox, around March 21, marks the Sun’s crossing of the equator on its way north. It crosses the equator again on its way south at the autumnal equinox around September 21.

But what is a year? This question is not quite so trivial as it may seem. A sidereal year is the period of the Earth’s annual revolution about the Sun and return to its original position relative to distant stars. Its duration is 365.25636 mean solar days. The tropical year is measured from one vernal equinox to the next. However, the reference point (Aries) moves during the year because of precession, and consequently the tropical year is shorter at 365.24219 days. The anomalistic year is measured from perihelion to perihelion, and is 365.25964 days. This long period is due to a slow eastward drift of the Earth’s perihelion.

In the next chapter it is shown that according to Newton’s theory of planetary motion, the period. T, and the semi-major axis, a, of an orbit are related by

where G is the universal gravitation constant and M is the mass of the Sun. This equation applies to all the planets in the Solar System. For the Earth, a ≈ 1 astronomical unit, and T = 1 year, so, in these units GM ≈ (2π)² cubic au per square year. The Gaussian year of 365.25690 days is defined by setting a = 1 in (3.1). (The au has now been defined in km, so that the semi-major axis of the Earth’s orbit is only approximately 1au, and changes slowly with time.)

1.4 The modern calendar

In 1267 an English Franciscan friar, Roger Bacon, drew the attention of Pope Clement IV to a problem with the Julian calendar. The problem was that the calendar year of 365¼ days was almost 11¼ minutes longer than the tropical year which governs the rhythm of the seasons. This meant that the calendar would count an extra day every 128 years. By the 13th century, the accumulated discrepancy amounted to about eight days. This meant that the vernal equinox, supposedly on 21 March, had actually occurred more than a week previously. This was quite a serious matter for the Church as it affected the date of the celebration of Easter. However, the Pope, who was sympathetic to Bacon’s ideas, abruptly died and nothing was done about the calendar until the papacy of Gregory XIII some 300 years later. What was done in 1582 was astonishing, especially to the devout Christians of Europe. A papal bull was issued stating that the day following Thursday October 4 would be Friday October 15. Ten dates were simply abolished! Moreover, an adjustment was made to the rule governing leap years. This was that, henceforth, centenary years, except those which were multiples of 400, would not be leap years. Thus 1600 and 2000 would be leap years, but 1700, 1800 and 1900 would not. Four Gregorian centuries would therefore contain 146,097 days instead of the 146,100 days in four Julian centuries. This change had the effect of reducing the discrepancy between the calendar and the tropical year to 27 seconds, postponing any further need to reform the calendar for more than 3000 years (Reference 10).

In spite of this, the new calendar was not very readily adopted, especially in non-Catholic countries. It was adopted by Great Britain and its colonies in 1752, by Japan in 1873, and by China in 1949.

1.5 Julian Day numbers

Astronomy requires the accurate measurement of time, sometimes over very long periods. For this reason astronomers literally count the days. The idea is that every day is assigned a number, the Julian day number, JD. E.g. 2000 January 1 is JD = 2451545.0. This integral value refers to 12 noon on the day. JD for the previous midnight is 0.5 less and that for the next midnight 0.5 more. Extremely long periods are counted not in years, but in Julian centuries of 36525 days or Julian millennia of 365250 days. Figure 1.2 presents FORTRAN 77 source codes for two sub-segments which are important inclusions in any programme for calculating planetary ephemerides. The algorithm giving JD for any calendar date is 1.2(a), while the algorithm for determining the calendar date corresponding to a given JD is 1.2(b). The scheme assigns a unique JD number to any valid date. The algorithm in Fig. 1.2(a) will also assign a JD number to an obviously invalid date, such as July 52, but not a unique one; it will always coincide with the JD for a valid date, in this case August 21. Similarly each of the 10 dates in October 1582 abolished by

Figure 1.2a JD–2451600.5 for a given date. (Fortran source code)

Figure 1.2(b) The calendar date for a given JD. (Fortran source code: both this and 1.2(a) are based on algorithms given in Reference 1, Chapter 7)

Pope Gregory shares its JD number with the valid Gregorian calendar date obtained from it by adding 10 days. In valid leap years, February 29 and March 1 have distinct JDs differing by 1. In all other years, they are identical. Negative dates are allowable and often used in ephemeris tables, e.g. 2013, January–8 has the same JD as 2012, December 23. Thus the scheme counts all the valid dates from JD = 0.0 (noon on New Year’s Day 4713 BCE), through the extrapolated Julian calendar up to its last day, 1582 October 4, then on through the Gregorian calendar, beyond the present day and into the distant future if need be. By considering negative JD numbers the count can be continued into the even more remote past, the dates being assigned by extending the Julian calendar backwards from 4713 BCE. (N.B. Care is needed when entering BCE dates into a computer programme; e.g. Julius Caesar’s first arrival in this country, 55 BCE must be entered as −54. His next visit in 54 BCE is −53.)

By noting the remainder when JD (at noon) is divided by 7, we can infer the day of the week corresponding to the given JD. No remainder = ‘Monday’, remainder 1 = ‘Tuesday’, and so on up to remainder 6 = ‘Sunday’. E.g. New Year’s Day 2000, has JD = 2451545, which leaves remainder 5 when divided by 7. It was therefore a Saturday.

1.6 Sidereal time

Figure 1.3 is a crude representation of the Earth rotating on its polar axis and revolving around the Sun, S. Two positions of the Earth are shown: the first in June when the star E1, which is close to the intersection of the ecliptic with the local meridian of the observer at A', belongs to the constellation Sagittarius. The second is a month later when the Earth has moved a further 30° around its orbit. The star E2 which the observer A' sees on his local meridian at midnight belongs to the constellation Capricorn, 30° east of E1. During the month which has elapsed, the Earth has rotated on its polar axis 30 times and the observer at A has seen the Sun rise in the east, transit his local meridian and set in the west 30 times. Similarly, the observer at A' has seen the star E1 rise, transit and set 30 times. However, it is currently doing these things two hours earlier than it did in June, because it is 30° west of where it was in relation to the Sun a month earlier. From this it is evident that, in crossing the sky each day, the stars follow a schedule which differs from that of the Sun. As the Earth moves around its orbit and the Sun apparently moves eastwards amongst the stars, the stars move westwards relative to the Sun. Every month the stars move a further 30° west, so that after 12 months this amounts to 360°, or a complete circuit of the sky. This implies that, while to an Earth-bound observer the Sun apparently revolves around the Earth 365¼ times in a year, the stars must do this 366¼ times.

Figure 1.3 Solar time and sidereal time.

Evidently the stellar clock ticks a little faster than the solar one. Time reckoned by the stars is called sidereal time. It follows from the argument above that 1 solar day is equal to 1 + 1 / Y sidereal days where Y is the number of days in a year. This means that one sidereal day should be close to 24h × 1461/1465 = 23h 56m 4s. Greenwich sidereal time (GST) is defined as the Greenwich Hour Angle (converted from degrees to hours), not of a particular star, but the point in the sky occupied by the Sun at the March equinox. This point is known as the first point of Aries or just Aries and is denoted by the symbol . It was in Aries