Configurations and synodic periods

Builds on: Sections 1.6, 2.1 and 3.1
Reading time: 35 min
Exercises: 4
Olympiad problems: 3

On 16 January 2025 Mars was at opposition. It stood opposite the Sun in the sky, rose at sunset, was highest at midnight and shone more brightly than any star except Sirius. The next opposition of Mars came on 19 February 2027, 764 days later, and in between Mars spent months lost in the glare of the Sun. Venus has a rhythm of its own. For many months it is the evening star, bright in the west after sunset, and then within a couple of weeks it moves across to the morning sky before dawn. In this section we see where these rhythms come from. They are set by two planets going round the Sun at different speeds, with Earth one of them.

6.1.1Inferior and superior planets

Before you readMake a guess first

Can Venus ever be seen high in the south at midnight from a northern country?

The planets whose orbits lie inside Earth's, Mercury and Venus, are called inferior planets. Those whose orbits lie outside it, from Mars outwards, are called superior planets. The names only say on which side of Earth's orbit the planet goes round, and nothing about the planets themselves.

From Earth we see a planet in some direction compared with the Sun. The angle between the two directions is the planet's elongation. We call it east when the planet is east of the Sun along the ecliptic, and west when it is west of it. A planet east of the Sun sets after the Sun, so it is seen in the evening. A planet west of the Sun rises before the Sun, so it is seen in the morning.

A few arrangements of the Sun, Earth and a planet have names, because they are the moments that observers and problems care about. All planets have a conjunction, when the planet is in the same direction as the Sun and its elongation is zero. An inferior planet has two kinds. At inferior conjunction it is between Earth and the Sun, and at superior conjunction it is on the far side of the Sun. Between them it reaches its greatest elongation, east or west, when it is as far from the Sun in our sky as it can be. A superior planet can never come between Earth and the Sun. Instead it has an opposition, when it is opposite the Sun with an elongation of 180°, and two moments of quadrature, when it is 90° east or west of the Sun.

Figure 6.1The configurations of a planet

Planet

Time

Watch

Elongation
Next configuration
Motion against the stars
Turns made round the Sun
Synodic period

The figure shows the Sun S, Earth E and a planet P on circular orbits, seen from above Earth's north pole, so that both go round anticlockwise. The heavy line is our line of sight to the planet, and the small arc at E marks the elongation between it and the direction of the Sun. Day 0 is an inferior conjunction for Mercury and Venus and an opposition for Mars, and the dashed line keeps the direction of that line-up. The slider runs through one synodic period, and the table gives the elongation, the next named configuration, the direction in which the planet moves against the stars and how many turns each body has made round the Sun. At the end of the period the new line-up is drawn heavy and dashed, with the angle by which it has moved on from the first. The figure can be set to , , , or .

At greatest elongation the line of sight from Earth just touches the planet's orbit. A line that touches a circle is at right angles to the radius at that point, so the triangle formed by the Sun, the planet and Earth has a right angle at the planet. Its hypotenuse is the distance from Earth to the Sun, and the side opposite the elongation is the radius of the planet's orbit.

ResultThe greatest elongation of an inferior planet(6.1)

\sin ε_{max} = \frac{a}{a_{\oplus}}

$ε_{max}$	the greatest elongation
$a$	the radius of the planet's orbit
$a_{\oplus}$	the radius of Earth's orbit, 1 au

This holds for circular orbits. For Venus, with

a = 0.723

au, it gives 46.3°. Mercury's orbit is so eccentric that its real greatest elongations range from about 18° to 28°, around the circular value.

Quick checkMercury's greatest elongation

Mercury's orbit has a radius of 0.387 au. What is its greatest elongation, if both orbits are taken as circles?

6.1.2The synodic period

Before you readMake a guess first

Mars takes 1.88 years to go round the Sun. How often is Mars at opposition?

The time a planet takes to go once round the Sun, measured against the distant stars, is its sidereal period. The time between two repeats of the same configuration, such as two oppositions of Mars, is its synodic period. The two differ because Earth moves too.

The picture is that of two runners on circular tracks round the same centre. The inner runner has the shorter lap and gains on the outer one all the time. They are lined up again when the inner runner has gained one whole lap. Earth turns through $360 ° / P_{\oplus}$ each day, and a planet of sidereal period $P$ through $360 ° / P$ , so for a superior planet Earth gains $360 ° (1 / P_{\oplus} - 1 / P)$ each day. It takes one synodic period $S$ to gain 360°.

ResultThe synodic period(6.2)

\frac{1}{S} = | \frac{1}{P} - \frac{1}{P_{\oplus}} |

$S$	the synodic period, the time between repeats of the same configuration
$P$	the sidereal period of the planet
$P_{\oplus}$	the sidereal period of Earth, 365.26 days

The rule holds for any two bodies going round the same centre in the same direction, with

P_{\oplus}

replaced by the period of the second body. It is exact for circular orbits with steady speeds, and gives the average interval when the orbits are ellipses.

For Mars, with $P = 686.98$ days, the synodic period is 780 days, or 2.135 years. Venus has a synodic period of 583.9 days and Mercury of 115.9 days. The further out a superior planet is, the more slowly it moves, and the closer its synodic period comes to one year. Jupiter's is 398.9 days and Saturn's 378.1 days, so these planets come to opposition a few weeks later each year.

The same rule gives the month of the phases. The Moon goes round Earth once against the stars in 27.32 days, its sidereal month. In that time Earth has carried the Moon 26.9° round the Sun, so the Moon needs about two more days to catch up with the direction of the Sun and be new again. With $P = 27.3217$ days for the Moon and $P_{\oplus} = 365.26$ days, the synodic month is 29.53 days. The difference between the sidereal day and the solar day in (2.1) is the same rule once more, with Earth's spin in place of the Moon's orbit.

Quick checkOppositions of Jupiter

Jupiter's sidereal period is 4332.6 days. How many days pass between two oppositions of Jupiter?

days

Real orbits are ellipses, so real synodic periods are not all equal. From 1900 to 2050 the intervals between oppositions of Mars range from 764 to 811 days, with an average of 779.6 days. The interval from 2025 to 2027 is within a day of the shortest of them all, because Mars was then near the far end of its orbit, where it moves most slowly and Earth catches it up soonest.

Example 6.1From opposition to quadrature

How long after an opposition of Mars does it reach eastern quadrature? Take circular orbits of radius 1 au and 1.524 au.

Solution

At quadrature the angle at Earth between the Sun and Mars is 90°. The triangle formed by the Sun, Earth and Mars has a right angle at Earth, so the angle $φ$ at the Sun between Earth and Mars has $\cos φ = 1 / 1.524$ , which gives $φ = 49.0 °$ .

At opposition this angle was zero, and it grows by 360° in one synodic period of 780 days. It reaches 49.0° after

t = \frac{49.0 °}{360 °} \times 780 days = 106 days .

AnswerMars reaches eastern quadrature about 106 days, three and a half months, after opposition. In the figure, Earth is ahead of Mars at that point, so Mars appears 90° east of the Sun and is due south at sunset.

6.1.3Where the next configuration falls

A synodic period is not a whole number of years, so each repeat of a configuration comes at a different place along the ecliptic and a different time of year. The line-up of the Sun, Earth and an outer planet points wherever the outer planet has got to. In one synodic period the outer planet moves round by $360 ° \times S / P$ , so that is how far round the Sun each repeat falls from the last.

For Mars this is 408.7°, a whole turn and 48.7° more. Each opposition of Mars is therefore about 49° further east along the ecliptic than the one before, and about seven weeks later in the year, since the Sun takes that long to move 49°. Jupiter's oppositions move on by 33.1° each time, about one constellation of the zodiac each year.

For Venus the outer body is Earth, which moves on by 215.5° beyond a whole turn in each synodic period. Five synodic periods of Venus take 7.99 years. After five inferior conjunctions, then, Earth has gone round almost exactly eight times, and the sixth inferior conjunction falls only 2.5° away from the first, on nearly the same date. The evening and morning appearances of Venus repeat on an eight-year cycle for this reason.

Example 6.2Oppositions of Saturnfinish it

Saturn's sidereal period is 10 759 days and its synodic period is 378.1 days. By how many days does the date of opposition move later each year?

The start

Each opposition is one synodic period after the last. In that time Saturn moves on by $360 ° \times 378.1 / 10 759 = 12.7 °$ , and the Sun has to move that far beyond a full year to be opposite Saturn again.

Your stepNow find how many days later in the year each opposition comes.

days

Common errorsSection 6.1

The synodic period comes from subtracting the frequencies $1 / P$ , not the periods. Subtracting 365 days from 687 days gives nothing useful.
The absolute value in (6.2) matters. For an inferior planet $1 / P$ is the larger term, and for a superior planet $1 / P_{\oplus}$ is, but the synodic period is positive in both cases.
A superior planet is east of the Sun after opposition and west of it before. After opposition Earth moves ahead, so the planet falls back towards the evening sky and reaches eastern quadrature first.
The formula for greatest elongation takes the sine of the angle, not the tangent. The right angle is at the planet, where the line of sight touches its orbit, not at the Sun.
Sidereal periods must be used in the formula. Using the synodic month of 29.53 days as the Moon's orbital period, for example, puts the error of two days into every later step.

ExercisesSection 6.14 questions

6.1.1
Venus has a sidereal period of 224.70 days. How many months of 30.44 days pass between two inferior conjunctions?
months
6.1.2
On a certain evening a superior planet is due south at sunset. Which configuration is it in?
6.1.3
The asteroid Ceres goes round the Sun on a nearly circular orbit of radius 2.77 au. How many days pass between two of its oppositions? Take a year as 365.26 days.
days
6.1.4
Which of these planets could be seen due south at midnight from the northern hemisphere?

Olympiad problemsPast IOAA problems, restated3 problems

2010The next great opposition of Mars10 points

Adapted from IOAA 2010 (Beijing), theory problem 11. We have restated it in our own words and kept the original data. The solution was last checked on 23 October 2026.

Mars reached a great opposition, an opposition at which it came especially close to Earth, at 17h 56m UT on 28 August 2003. The next great opposition came in 2018. Estimate its date. The semi-major axis of Mars's orbit is 1.524 au. Take both orbits to be circles.

I solved it without the solution

AnswerOur estimate puts the great opposition on 7 August 2018. The real opposition came 11 days earlier, on 27 July 2018. Mars's orbit is an ellipse, so its speed changes round the orbit, and the seven real intervals did not add up to seven average synodic periods.

2023An opposition of Neptune at the equinox5 points

Adapted from IOAA 2023 (Katowice), theory problem 1. We have restated it in our own words and kept the original data. The solution was last checked on 23 October 2026.

Neptune was at opposition on 21 September 2024. In which year was Neptune last at opposition near the time of the March equinox? Take the orbits of Earth and Neptune as circles, and the semi-major axis of Neptune's orbit as 30.070 au.

I solved it without the solution

AnswerThe opposition came about 82.4 years before late September 2024, near year 1942.3, so it was the opposition in the spring of 1942. The official solution accepts 1941 or 1942 under these assumptions. With the real, slightly eccentric orbits the closest opposition to the equinox was in 1943.

2017The great conjunctions and the Star of Bethlehem20 points

Adapted from IOAA 2017 (Phuket), theory problem 8. We have restated it in our own words and kept the original data. The solution was last checked on 23 October 2026.

A great conjunction is a conjunction of Jupiter and Saturn as seen from Earth. Take the orbits of Jupiter and Saturn as circles in the plane of the ecliptic, with sidereal periods of 11.86 and 29.45 years. The time between great conjunctions seen from Earth varies a little, but on average it is the same as for an observer at the centre of the Solar System. Ignore precession throughout.

(a) Find the average time between great conjunctions, in years, and the average angle round the Sun between one great conjunction and the next. [6 points]

(b) A great conjunction took place on 21 December 2020, 30.3° east of the Sun. In which constellation was it? [2 points]

In 1606 Johannes Kepler worked out that a great conjunction can happen three times in one year, because both planets loop back in retrograde motion, and that this happened in 7 BC. It may be the event remembered as the Star of Bethlehem.

(d) In which constellation was the Sun at the second of the three conjunctions of 7 BC? [4 points]

I solved it without the solution

Answer(a) The great conjunctions come every 19.86 years on average, each 242.7° further round the Sun than the last. (b) The conjunction of 2020 was in Capricornus. (c) Those of 7 BC were in Pisces. (d) The Sun was in Virgo. The official solution allows Aries and Libra as well, since an average period carried back over two thousand years cannot place the conjunction to better than a few degrees.