Close Encounters of the Martian Kind and Other Astronomical Stories of 2003

Biography

Dr Michael McCabe is a Principal Lecturer in the newly created Department of Mathematics at the University of Portsmouth.  His special interests are the teaching of astronomy and the effective use of learning technology.  In 2001 he was awarded a HEFCE National Teaching Fellowship in recognition of his teaching excellence.

Abstract

The closest approach of Mars to Earth for thousands of years, a transit of Mercury, two lunar eclipses and a solar eclipse are among the astronomical events, which UK observers can look forward to in 2003.  Mathematical calculations, which have increased in precision over the centuries, have made such predictions possible.

An Eventful Year

2003 will be an eventful year for astronomy in our Solar System. At dawn on 31^st May in the north of Scotland there will be an annular eclipse of the Sun and a partial solar eclipse elsewhere in the UK.  On 16^th May and 9^th November there will be lunar eclipses and on 7^th May there will be a transit of the planet Mercury across the face of the Sun. 

 There will also be considerable attention focused upon the planet Mars. In June the Mars Express mission will be launched towards the red planet with the British-built Beagle 2 lander on board.  In December Beagle 2 will begin its search for life by analysing rocks on the surface of the planet. Furthermore, on 27^th August Mars will be closer to Earth than it has been for well over 6000 years.  The UK National Astronomy Week is being held from 23^rd to 30^th August to mark this event, which will allow good views of Mars and enable surface details to be picked out through a modest-sized telescope. 

A Closer Look at Mars

Detailed predictions of these events and planning of planetary missions would, of course, be impossible without mathematics.  Suppose we begin with Mars. Why is Mars coming so close and why is it so long since it was last at a comparable distance? 

Indeed does such an occurrence have any great significance?

 Earth and Mars orbit the Sun at distances of 1.0 and 1.52 Astronomical Units (AU) respectively. According to Kepler’s First Law their orbits are ellipses with the Sun at one focus.  The eccentricities of their orbits are 0.0167 and 0.0934 respectively, significantly larger for Mars, but in both cases it is surprisingly difficult to distinguish them visually from circles.  Figure 1 shows an accurate scale diagram of Mars orbit.

 Figure 1 Screen from Mathwise Astronomy (developed by the author)

 According to Kepler’s Third Law, the orbital period P, in years, is related to the radius, or more accurately the semi-major axis a of the orbit, in AU, by the relationship P² = a³ or P = a^3/2.

For Earth P = 1^3/2 = 1 year; for Mars P = 1.52^3/2 = 1.88 years. 

 Because its orbital period is shorter, the Earth overtakes Mars at regular time intervals when the two planets are said to be at opposition.  At opposition Mars is relatively close to the Earth (see Figure 2) and appears opposite the Sun in the sky.  Conveniently, this means that Mars is high in the sky during the middle of the night and therefore easier to observe when it is closer to Earth. How often then do such oppositions occur?

 Sidereal and Synodic Periods

Let S be the time between two successive oppositions, or identical configurations of Mars with respect to the Earth, known as the synodic period of Mars.  Let P be the orbital period of Mars around the Sun, the sidereal period of Mars, and E be the orbital period of the Earth around the Sun, i.e. 1 year.  

 The angular rate at which the Earth and Mars move around their orbits are 2p/E and 2p/P respectively.  During a time S, the Earth travels an angular distance of (2p/E)S.

At the same time, Mars travels an angular distance of (2p/P)S.  Mars though has travelled one less orbit than the Earth, as shown in Figure 2.

 Figure 2 Orbits of Earth and Mars Between Two Consecutive Oppositions

Hence,             (2p/P)S = (2p/E)S  - 2p

 => 1/P = 1/E – 1/S      =>  S = (1/E – 1/P)^-1 =  EP/(P – E)

 The synodic period, or period between oppositions, of Mars is

 S = (1 × 1.88) / (1.88 – 1) = 1.88/0.88 = 2.136 years or 780.3 days

Hence, there is an opposition of Mars approximately every 2 years and two months. For example, the last opposition of Mars occurred on June 13^th 2001 and the next will be on 27^th August 2003 as shown in Figure 3.

 Figure 3 Oppositions of Mars

If Mars comes close to Earth roughly every 2 years 2 months, why then will it be reaching its closest for over 6000 years in 2003? We need to dig a little deeper.

 The closeness of encounters with Mars at opposition needs to take into account the elliptical orbits of the planets, shown in Figure 4. The maximum and minimum distances of Mars from the Sun, its aphelion q and perihelion p, are given by

 q (Mars)  = a (1 + e) = 1.5235 (1 + 0.0935) =  1.666 AU

 p (Mars) = a (1 - e) = 1.5235 (1 - 0.0935) = 1.381 AU

where e = 0.0935 is the eccentricity and a = 1.5235AU is the semi-major axis of Mars orbit.

Similarly q(Earth) = 1.017 AU and p(Earth) = 0.983 AU  

 Theoretically, the closest possible value of the Earth-Mars distance is the difference between the perihelion of Mars and the aphelion of Earth

p (Mars) - q(Earth) = 1.381-1.017 =  0.364 AU.

At the moment this cannot occur, because the aphelion of Earth and the perihelion of Mars do not line up.  They are, in fact, roughly 57 degrees out of phase (Figure 4).

Partly as a consequence of this, the dates of closest approach are not necessarily identical to the dates of opposition, e.g. Mars actually has its closest approach on 27^th August 2002 and not 28^th August.

Figure 4  Relative Positions of Earth Aphelion and Mars Perihelion

Favourable Oppositions

Because the orbit of the Earth has such a low eccentricity, it can be considered as circular to a first approximation. Mars therefore comes especially close to Earth when it is simultaneously at opposition and near the perihelion of its elliptical orbit, a configuration known as a favourable opposition.  How regularly does this happen?

The interval between such favourable oppositions can be estimated by considering how often Mars and Earth complete approximately integral numbers of orbits at around the same time, i.e.  nP » mE = m years where n and m are integers. Using the approximate value for P =  1.88 years, then n = 8 and m = 15.04 » 15 years gives good alignment of the planets.  Figure 3 not only shows the years and the time of year that successive oppositions occur, it also shows how their position moves around a complete orbit approximately every 15 years.  For example, Mars was last at opposition in June 2001, but the last time it had a favourable opposition was in September 1988.  This is still a far cry from 6000 years ago.

Ultra Favourable Oppositions

The next step is to consider whether more accurate oppositions of Mars can occur over longer time intervals. A more accurate orbital period for Mars is

P = 686.980 days = 686.980/365.2564 = 1.8808158 years.

Once again we seek n such that  nP » m where m is an integer.

Table 1  Time Intervals nP between Ultra Favourable Oppositions of Mars

 The final column of the Table 1 shows the deviation D from an integral number of Mars orbits at opposition, i.e. the misalignment between Mars and Earth at favourable oppositions with the period nP.  At intervals of around 79 years the alignment of Mars and Earth when Mars is at perihelion is accurate to around 2 degrees and at an interval of 647 years the alignment is accurate to within a small fraction of a degree. Table 2 shows the dates of the 27 closest approaches of Mars in the period 4713 BC to 4000 AD, calculated using the astronomical computer package Sky Map Pro.  It also gives the time interval between these close approaches, the distance to Mars and the ranked order of the closeness to Earth.  

Table 2  Closest Approaches of Mars Between 4713 BC and 4000 AD (D. Harris)

 The first column of Table 2 gives the date of the close approach, which is approximately, but not necessarily exactly, the date of the perihelion of Mars.

 The second column shows the time interval between these close approaches, where it can be seen that the intervals of 47 and 79 years crop up regularly and correspond to the time intervals shown in Table 2.  The three close approaches of 1561, 2208 and 2855 shown in bold, are separated by 647 years, the interval between ultra favourable oppositions.

 The distances to Mars are shown in the third column and the ranking of closeness in the fourth column.  It can be seen that, while there are no closer approaches listed earlier than the opposition of 2003, there will be many subsequent occasions when Mars gets closer.  Furthermore, not all the closest approaches occur at the longest 647 year time interval. It is clear from all this that there is more to it than the simultaneous opposition and perihelion of Mars.

Orbital Precession

It was noted earlier that the perihelion of Mars and the aphelion of Earth, currently occurring in late August and early July respectively, are not aligned (Figure 4).  Allowance for the slightly elliptical orbit of the Earth and its closer approach to Mars at aphelion does not in itself make any significant difference to the intervals between close approaches.  It simply modifies the dates of closest approach, so that they do not coincide with the perihelion of Mars. The fact that the orbits of both Earth and Mars precess does though make a significant difference.

 Our annual calendar on Earth is not based upon the orbital period of the Earth around the Sun, the sidereal year, but rather the period from one equinox to the next, the tropical year.  Because of axial precession, the slow wobbling of the Earth’s spin axis like a top, the tropical year is over 20 minutes shorter than the sidereal year. There is yet another type of year, the anomalistic year, defined as the time taken from one perihelion passage to the next.  Because of orbital precession, the gradual rotation of the Earth’s orbit within its orbital plane (also known as the precession of the equinoxes), the anomalistic year, is 25 minutes longer than the tropical year. 

Figure 5  The Orbital Precession of Earth

Figure 5 shows the position of the Earth’s orbit at time intervals of around 1500 years.  The main cause of orbital precession is the gravitational interaction of other bodies in the Solar System, primarily the Moon and the planet Jupiter, with the Earth. 

 Since  1 year / 25 minutes = (365 x 24 x 60)/25 » 21,000

 the period of the orbital precession is approximately 21,000 years. It is interesting, but irrelevant here, to note that there is a very small “anomalous component” to orbital precession of about 5 arc seconds per century, which can be explained by General Relativity.

 Now the orbit of Mars also precesses, albeit at a rather slower rate, having a period of around 51,000 years.  Since the orbits precess in the same direction, the aphelion of the Earth “catches up” the perihelion of Mars every 30,000 years (Figure 6).  Based upon these figures, it will take another 5000 years or so for Mars to be able to reach its ultimate closest distance of p(Mars) - q(Earth) = 0.364 AU.  The distances shown  in Table 2 shows how the close approaches of Mars are gradually creeping towards this limiting value.

 Current publicity states that “Mars will be at its closest for over 6000 years” based upon computer calculations stretching back to 4713 BC.  (The significance of the date is that astronomers measure time arbitrarily from noon GMT on January 1^st 4713 BC, defined as Julian Day 1.  Computer programs work from this date forwards without complications of negative numbers, changes to calendars, leap years etc. Prior to this date inaccuracies tend to increase.)  In fact, the last time that there would have been comparable alignment of the Earth’s aphelion and Mars perihelion would have been around 30,000 – (2 x 5000) = 20,000 years ago, so it is probably safe to argue that Mars will be at its closest for 20,000 years!  The significance of this is more debatable, since Mars will be able to make marginally closer near approaches progressively for the next 5000 years (Table 2)!  Even so, it is highly recommended that the opportunity to view Mars next summer is not missed.

 It is worth noting that there are other effects caused by the gravitational pull of the Moon and planets, such as gradual changes in the eccentricity of planetary orbits. The fluctuations in the general trend of the Earth-Mars distance in table 2 can be attributed to planetary or lunar alignments, as well as the 1.8 degree inclination of the orbit of Mars to the ecliptic.

 Figure 6  Change in Date of Earth Aphelion and Mars Aphelion with Calendar Year

Solar Eclipses

The elliptical orbits of the Moon around the Earth and, to a lesser extent, the Earth around the Sun are important in understanding the 2003 solar eclipse.  Why are we not being treated to a total solar eclipse as in August 1999?  My favourite analogy is the beachball (Sun), pea (Earth), peppercorn (Moon) and length of a football pitch (Earth-Sun distance).  Imagine a three-foot diameter beach-ball in one goalmouth and a pea in the other.  Viewed from the pea, the peppercorn at a distance of one foot away can block the view of the beach-ball in the other goal. That’s an eclipse!

Consider the actual scale in more modern units:

 The Earth is at its perihelion and aphelion distances, p and q, from the Sun on July 4^th and December 4^th respectively. 

 q(Earth) = 1.521 x 10⁸ km = 1.017 AU            p(Earth) = 1.471 x 10⁸ km = 0.983 AU  

 The Moon is closest and furthest from Earth at its perigee and apogee respectively

q(Moon) = 4.055 x 10⁵ km                              p(Moon) = 3.633 x 10⁵ km

Diameter of the Moon                           d(Moon) = 3476 km

Diameter of the Sun                              d(Sun) = 1.392 x 10⁶ km

Ratio of diameters                                d(Sun)/d(Moon) = 400.4

Ratio of distances                                 362.8  £  D(Sun)/D(Moon) £ 418.7

 The similarity of these ratios means that the Sun and Moon have about the same angular diameter a.  To be more precise, a = 3437.75 d / D arc minutes.