Astronomy Answers: Conjunctions

Astronomy Answers
Conjunctions


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1. What is a conjunction of celestial bodies? ... 2. What does a conjunction mean? ... 3. How can you measure the closeness of a conjunction? ... 4. What conjunctions have been and are coming? ... 4.1. Mercury - Saturn ... 4.2. Other Combinations of Planets ... 5. How close together can the planets get? ... 6. Where are the planets now in the sky?

Once in a while there are rumors about upcoming conjunctions of "all" planets, which some people expect to have great consequences on Earth. For example, there was some (unwarranted) panic about the conjunction of some planets in May 2000. This essay explains about conjunctions of planets and other celestial bodies, and that those have no measurable influence on Earth, except for the tides that the Sun and Moon raise.

1. What is a conjunction of celestial bodies?

Fig. 1: Picture of Jupiter, Saturn and Pleiades
Fig. 1: Picture of Jupiter, Saturn and Pleiades
There is a conjunction of celestial bodies if those bodies are (temporarily) close together in the sky. The picture (taken by the author on 12 January 2001 with a digital camera) shows a conjunction of Jupiter and Saturn. Jupiter is the brightest "star" just right of center, and Saturn is in the lower right-hand corner. The small group of stars to the upper right of Jupiter are the Pleiades, and the brightest star at the left-hand side is Aldebaran. The two planets are about 7 degrees apart.

How close together must the celestial bodies be to be in conjunction? That depends on who you ask. If you think it is only a "real" conjunction if two planets are less than 10° apart, but your friend is satisfied already with 20°, then your friend will see more and longer lasting conjunctions than you will.

If there are more than two celestial bodies involved, then you must decide when all of them are in conjunction. Is that when they all fit within a circle of a certain diameter? Or when the distance between each pair of successive planets (from left to right) is less than a limit value? Or do you use still another measure?

It is clear that the meaning of the word "conjunction" is not very precise. By adjusting your definition you can find few, or instead many conjunctions.

2. What does a conjunction mean?

Only one conjunction in the sky has noticeable influence on Earth, and that is the conjunction of the Sun and the Moon. Such a conjunction happens whenever it is New Moon, and then the tidal forces of the Sun and Moon add up and we have spring tide with on average a larger difference between high and low tide than usual.

The distances and masses of the planets are such that they have no measurable tidal influence on Earth. This is clear from the following table, which lists the maximum tides due to the planets and the Sun, compared to the tides due to the Moon. The strength of the tide due to a planet or other body increases when the mass of the body increases, but decreases rapidly (as the third power) when the distance of the body increases. The tides due to the Moon are more important than the tides due to the Sun because the much smaller distance of the Moon outweighs the much greater mass of the Sun.

Table 1: Tides on Earth Due to Planets

Name Mass Distance Tides
Moon 0.0123 0.00257 1
Sun 332946 1 0.46
Person 70 kg 1 km 0.000'054
Venus 0.815 0.28 0.000'051
Jupiter 318 4.20 0.000'005'9
Mars 0.107 0.52 0.000'001'1
Mercury 0.0553 0.62 0.000'000'32
Saturn 95.2 8.53 0.000'000'21
Uranus 14.5 18.18 0.000'000'003'3
Neptune 17.2 29.05 0.000'000'000'97
Pluto 0.00256 29 0.000'000'000'000'14

The column "Name" lists the name of the (celestial) body. The column "Mass" displays the mass; for the planets, Sun, and Moon these are compared to the mass of the Earth. The column "Distance" shows the typical least distance from the Earth; for the planets, Sun, and Moon these are measured in Astronomical Units. The column "Tides" provides the magnitude of the tides due to that body when it is at the indicated distance, compared to the tides due to the Moon. Usually, the planet is further away than the minimum distance listed in the table, so usually the tides due to the planets are even smaller than those listed in the table.

It follows from the table that the tides on Earth due to the Sun are about half as strong as the tides due to the Moon, and that the tides due to all other planets combined are at their greatest still some 15,000 times smaller than the tides due to the Moon. If the difference between high and low tides due to the Moon is 1.5 m (3 ft) somewhere, then the difference due to the Sun is about 0.7 m (1.5 ft), and the difference due to all other planets combined is at most about 0.1 mm (1/250th of an inch): so small that it cannot even be measured.

The table also lists the tides due to a person of 70 kg (155 lb) at 1 km (0.6 mi) distance: those tides are even larger than the tides due to any planet! And every time that the distance of that person is divided in two, the tides increase eightfold. The tidal forces due to a person at about 40 m (120 ft) is comparable to the tidal forces due to the Moon. The distribution of all people, cars, buildings, and other heavy bodies within about 1 km (or 1 mi) from you has more influence on you than the configuration of the planets.

Of the four fundamental forces in the Universe, only gravity (and the associated tidal forces) is effective at great distances. If the tidal forces of the planets are negligible on Earth, then the other fundamental forces due to the planets are even more negligible on Earth. In short, conjunctions of planets sometimes provide pretty sights in the sky, but are otherwise of no importance.

3. How can you measure the closeness of a conjunction?

There is no clear border between having a conjunction and not having a conjunction, so it is more useful to use a measure that indicates how close the conjunction is at any moment. With such a measure, you can also effectively compare different conjunctions.

An obvious measure for the closeness of a conjunction of planets is the diameter of the smallest circle that encloses all of the planets, but that circle can usually only be found after a tedious search, and does not depend on the distribution of the planets within the circle.

A better measure for calculating is what I call the conjunction spread. The calculation of the conjunction spread is fairly easy and requires no searching, and this measure changes whenever any one planet's location changes.

You calculate the conjunction spread as follows: Determine for each planet that is included the vector of length 1 that points from Earth to that planet. Call the length of the average of all of those vectors \( r \). The conjunction spread \( w \) in degrees is then equal to

\begin{equation} w = \sqrt{−26262.45\ln(r)} \end{equation}

with \( \ln \) the natural logarithm. For planets distributed randomly across the ecliptic, with standard deviation \( s \) in the ecliptic longitude, \( w \) equals twice \( s \). For two planets that are close together, their conjunction spread is almost equal to their distance from one another. (The conjunction spread then overestimates the distance by less than one percent for distances less than 40 degrees.)

4. What conjunctions have been and are coming?

4.1. Mercury - Saturn

I've calculated the conjunction spread (as seen from Earth) for the planets Mercury through Saturn, which can be seen with the unaided eye, for a period of eleven million days between the years −13200 and 17191. To calculate the positions of the planets, I used the DE431 ephemeris of NASA JPL. The conjunction spread during this period shows periodic behavior with main periods of 378.09, 398.88, and 779.94 days, corresponding to the synodical periods of the Earth with Saturn, Jupiter, and Mars. The least conjunction spread is 2.7°, the greatest is 428.3°, and the average is 127.8°.

The next table shows for a few values of the conjunction spread during which fraction of time the conjunction spread of Mercury through Saturn (as seen from Earth) is less than or equal to that value.

Table 2: Conjunction Spread Distribution

spread (°) 11.2 19.8 35.3 55.5 69.1 125.2
fraction 1/10'000 1/1000 1/100 1/20 1/10 1/2

For example, the conjunction spread is smaller than 11.2° during only one ten thousandth of the time, and the conjunction spread is less than 125.2° during half of the time (and greater than that during the other half of the time).

Fig. 2: Diagram of the Chance for Conjunctions
Fig. 2: Diagram of the Chance for Conjunctions
Figure 2 also shows during which fraction of the time the conjunction spread of Mercury - Saturn is less than certain values.

The conjunction spread \( w \) is shown along the horizontal axis, measured in degrees. The vertical axis measures the chance (1 = everything) that the conjunction spread at a randomly selected moment is not more than the value displayed along the horizontal axis. For example, if you go up straight from the 10 on the horizontal axis until you hit the solid line and then go left until you hit the edge of the graph, then you end up at about 0.0001, which means that the part of the time during which the conjunction spread is 10° or less is equal to about 0.0001 or 0.01% or one part in ten thousand.

This diagram is a so-called double logarithmic plot. Short and long tick marks are indicated along the horizontal and vertical axes. Each next long tick mark represents a value that is ten times (an order of magnitude) greater than the previous one, as the associated numbers show. To get the values of the short tick marks you should multiply the value of the next left or lower long tick mark by 2, 3, 4 through 9. Then comes another long tick mark which represents 10 times as much as the previous long tick mark. The first couple of values associated with the long and short tick marks starting at the left margin of the diagram are: 2 (left margin, y axis), 3 through 9 (short), then 10 (long), 20 (short), 30 through 90 (in steps of 10), then 100 (long), 200 (short), and so on.

The dashes line shows the results of an approximation formula, equal to

\begin{equation} P(≤ w) ≈ 6.5×10^{−9}w^{4} \end{equation}

Fig. 3: Conjunction Spread Diagram
Fig. 3: Conjunction Spread Diagram
Figure 3 shows the conjunction spread for the years 1999 through 2004. There was a close conjunction during a few weeks around 11 May 2000. The conjunction spread then dropped to 15.1°, which means that the planets were spread over about 15 degrees in the sky then. If we call it a conjunction every time that the conjunction spread reaches a minimum (and so starts going up again), then the investigated period of 30390 years contains 237 conjunctions at least as close as that of May 2000, so such a conjunction happens on average about 8 times per 1000 years (without a clear period of repetition).

The next reasonably close conjunction occurred in May 2002, with a conjunction spread of 23°. Conjunctions that are as close or closer than that one occur about 27 times per 1000 years during the investigated period (without a clear period of repetition).

Fig. 4: Conjunction Intervals Diagram
Fig. 4: Conjunction Intervals Diagram
In Figure 4 you can see how much time there is, on average, between two successive conjunctions (local minimums in the conjunction spread) that are closer than a selected value. For example, a conjunction with a conjunction spread of at most 10° occurs on average once per 375 years, and a conjunction spread of at most 30° happens on average once every 15 years. The correspondence between the conjunction spread \( w \) and the average time interval \( t \) is reasonably approximatd by

\begin{equation} w ≈ 75 (t - 0.35)^{−0.33} \end{equation}

Below is a table with information about the top 30 of closest conjunctions (with the smallest conjunction spreads) of Mercury through Saturn during the years −13200 through 17191:

Table 3: Closest Conjunctions of Mercury - Saturn from −13200 to 17191

JD a m d w c r
−2241219.7 −10849 11 16.8 7.4 +15.6 20
−2052558.3 −10332 5 27.2 7.8 +29.3 27
−1486894.6 −8783 2 7.9 7.7 −11.0 23
−1298238.9 −8267 8 13.6 5.5 +2.0 9
−652450.8 −6499 9 8.7 8.1 −0.3 30
−355250.0 −5685 5 19.5 5.9 −11.9 10
−267905.2 −5446 7 8.3 7.2 −9.8 17
−217299.0 −5307 1 25.5 6.9 +23.1 14
−43351.0 −4831 4 24.5 6.6 +10.3 13
725682.4 −2726 10 23.9 7.8 +7.4 28
1008145.0 −1952 2 25.5 2.9 −27.5 2
1334770.0 −1058 5 27.5 5.1 +24.8 8
1653935.5 −184 3 25.0 6.5 −29.5 11
1668670.1 −144 7 27.6 7.8 −4.1 26
1842597.5 332 10 4.0 7.2 −13.3 18
1980561.6 710 6 26.1 5.1 +21.1 7
2154504.8 1186 9 18.3 6.9 +4.9 15
2466405.9 2040 9 8.4 7.5 +24.5 22
3112193.8 3808 10 18.3 7.8 +22.4 25
3532625.9 4959 11 25.4 7.3 +19.5 19
3829810.8 5773 7 25.3 7.9 +11.4 29
3866518.1 5874 1 24.6 2.7 −7.8 1
4178415.2 6728 1 5.7 3.0 +15.4 3
4584136.9 7838 11 3.4 5.1 −15.8 6
4671494.5 8078 1 6.0 7.1 −21.7 16
5229924.5 9606 12 12.0 4.1 −17.1 4
5853731.9 11314 11 15.4 6.6 +20.3 12
6781985.2 13856 5 5.7 7.4 −17.4 21
7057952.5 14611 12 2.0 7.7 +23.1 24
7739686.1 16478 6 9.6 5.0 −4.7 5

The column marked "JD" shows the Julian day number. Column "a" (annum) contains the number of the year in astronomical reckoning (which recognizes a year 0; year −2 corresponds to 3 BC). Columns "m" and "d" list the month number (January = 1, and so on) and the day number. The dates are given in the Julian calendar for years up to AD 1582, and in the Gregorian calendar for later years. Column "w" lists the smallest conjunction spread for the conjunction (in degrees). Column "c" shows the location of the center of the group of planets in the sky, relative to the Sun (in degrees). A positive number for "c" means that (most of) the planets are East of the Sun and therefore visible in the evening. A negative number means that (most of) the planets are West of the Sun and therefore visible in the morning (before sunrise). Column shows "r" the rank of the conjunction in this list (number 1 is the closest).

The narrowest conjunction of Mercury through Saturn during the investigated period will occur in January 5874, when the conjunction spread will be only 2.7°. The narrowest so far (since the beginning of the period) occurred in February −1952, when the conjunction spread was 2.9°. The next future conjunction from the top 30 of the investigated period comes in September 2040, when the conjunction spread will be 7.5°. The last top 30 conjunction happened in September 1186, when the conjunction spread was 6.9°.

The conjunction of May 2000 happened too close to the Sun to be well visible, with some planets close and East of the Sun, and the others close and West of the Sun. In that respect, the conjunction of May 2002 was better, and the conjunction of September 2040 willl be better, with the planets on average 29 and 24° East of the Sun (and so visible in the evening).

Here is a table similar to the previous one, but showing the top-30 of the period from 1 January 1000 through 1 January 3000.

Table 4: Closest Conjunctions of Mercury - Saturn from 1000 to 3000

JD a m d w c r
2089087.7 1007 8 12.2 14.6 +1.3 15
2118557.5 1088 4 18.0 17.3 +28.5 26
2125795.1 1108 2 10.6 13.8 −24.8 13
2154504.8 1186 9 18.3 6.9 +4.9 1
2190392.5 1284 12 20.0 13.1 +11.5 10
2277747.2 1524 2 18.7 9.1 +6.4 4
2292478.5 1564 6 19.0 13.7 +31.9 12
2299724.5 1584 5 1.0 14.6 −30.5 16
2314456.7 1624 8 31.2 10.3 −5.4 6
2328436.9 1662 12 10.4 17.2 −9.0 25
2386276.3 1821 4 20.8 13.2 −17.5 11
2437702.0 1962 2 6.5 14.7 −5.0 18
2451675.9 2000 5 11.4 15.1 +1.2 20
2466405.9 2040 9 8.4 7.5 +24.5 2
2473593.8 2060 5 14.3 19.4 +9.9 30
2473648.6 2060 7 8.1 17.6 −28.9 28
2488384.9 2100 11 12.4 12.5 −15.4 9
2560214.2 2297 7 11.7 10.2 −23.2 5
2560946.7 2299 7 14.2 17.3 +7.3 27
2574942.6 2337 11 8.1 16.6 −3.3 23
2611639.2 2438 4 28.7 14.2 −15.7 14
2626352.7 2478 8 10.2 11.1 +16.9 7
2640335.0 2516 11 21.5 16.6 +13.8 24
2698166.5 2675 3 25.0 12.2 +11.0 8
2712899.6 2715 7 27.1 14.8 +34.0 19
2734875.0 2775 9 25.5 15.8 −0.6 21
2748869.4 2814 1 17.9 14.7 −14.8 17
2749605.5 2816 1 24.0 16.1 +13.4 22
2771584.1 2876 3 27.6 19.3 −26.5 29
2800291.9 2954 11 2.4 8.4 +1.6 3

4.2. Other Combinations of Planets

The below table shows some statistics for different combinations of planets. For convenience I've included the previously discussed combination (Mercury through Saturn).

μ min 30/30 −1/30 +1/30
Venus - Saturn 136.8 16478-06-05 (0.9) 4.0 1524-02-10 (2.6) 2378-02-03 (2.4)
Mercury - Saturn 127.8 5874-01-24 (2.7) 8.1 1186-09-18 (6.9) 2040-09-08 (7.5)
Mercury - Neptune 153.5 15367-08-16 (13.1) 24.8 −917-03-17 (22.5) 3211-08-01 (22.7)
Mercury - Pluto 161.8 −5544-02-15 (23.2) 38.8 947-06-13 (30.0) 2854-03-22 (37.1)

Column "μ" shows the average conjunction spread (in degrees) for the entire interval between years −13200 and 17191. Column "min" shows the smallest conjunction spread (between parentheses, in degrees) and the corresponding date (year-month-day). Column "30/30" shows the conjunction spread of number 30 in the top 30 of smallest conjunction spreads in the interval. Column "−1/30" shows the most recent conjunction from the top 30. Column "+1/30" shows the next conjunction from the top 30.

The more celestial objects are included in the calculations, the wider the conjunctions become. With 4 celestial objects the smallest conjunction spread is 0.9°. With 7 objects it is 13.1°, and with 8 objects it is 23.2°.

5. How close together can the planets get?

In movies (such as Lara Croft: Tomb Raider from 2001) they sometimes show conjunctions of all (or at least many) planets where these planets line up exactly, or at least appear so close together in the sky that you can see them all as big disks close together through a powerful telescope, but in reality such an alignment never happens. We saw earlier that the smallest conjunction spread of Mercury - Saturn (as seen from Earth) between the years −13200 and 17191 is 2.7°, which is about 5 times the apparent size of the Moon, and 180 times as large as the apparent diameter of Jupiter in the sky.

Fig. 5: Planet Orbits
Fig. 5: Planet Orbits
The planets follow fixed orbits around the Sun, and cannot appear just anywhere in the sky. Figure 5 shows the orbits of all planets on 1 January (in ecliptic coordinates) in the sky. The location of the Sun is indicated by the small square. The planets cannot appear in other places in the sky on that date.

Fig. 6: Diagram of the Closest Possible Conjunctions
Fig. 6: Diagram of the Closest Possible Conjunctions
By shifting the planets freely along their orbits we can find the closest conjunction that is possible in principle at any given day of the year. Figure 6 shows the results of a search for the closest possible conjunctions. The numbers along the horizontal axis show the beginning of the corresponding months; for example, 2 = the beginning of February. The vertical axis shows the smallest conjunction spread that I found, measured in degrees. For each planet, the orbit was taken from the orbital period that started on 1 January 2000. (The results for planetary orbits from 1 January 3000 are virtually the same: the standard deviation of the difference is only 0.002, and probably mostly due to the search algorithm.)

The smallest possible conjunction spreads (of Mercury - Saturn) is never greater than 1.21° (as is approached on 23 May and 26 November), and is never smaller than 0.30° (as is approached on 13 March near ecliptic coordinates 325°, −1° and elongation 29° east, and on 3 September near ecliptic coordinates 145°, +1° and elongation 18° east). These closest possible conjunctions always occur at least 6° and at most 29° from the Sun. If such conjunctions happen between about 10 December and 19 February or between about 9 June and 15 August, then they happen east of the Sun (so they are visible after sunset), and otherwise west of the Sun (so they are visible before sunrise).

The best visible of the closest conjunctions of Mercury through Saturn would happen near a 13 March at 29° east of the Sun, with a conjunction spread of 0.3°. The planets would then appear in the sky strung out along a line with a length of about 0.4°, which is only slightly less than the apparent diameter of the Moon, but is still 40 times greater than the apparent diameter of Jupiter, which would then appear biggest of the planets. So, even in the most favorable case (which has not occurred during the last 15000 years, and will not occur during the coming 15000 years, either) the planets are still far apart, compared to their apparent sizes.

6. Where are the planets now in the sky?

Through the Planet Positions Page you can find diagrams of the positions of the planets relative to the Sun, as seen from Earth, for some years before and after the year 2000. The diagrams below show the positions of the planets Mercury through Neptune for the years 2000 through 2003 and for 2040 and 2041, relative to the Sun. Find the desired time of the year on the horizontal axis, and then go straight up until you cross the line of the planet of interest. Then go straight to the left to find the associated time on the vertical axis. That number is the time difference between the planet and the Sun, measured in hours: the planet is due South that many hours earlier (for a negative number) or later (for a positive number) than the Sun, and the rising and setting of the planet are also approximately that much sooner or later than those of the Sun.

If a planet is in the top part of the diagram, then it is visible after sunset. If the planet is in the bottom part of the diagram, then it is visible before sunrise. If the planet is close to the upper or lower edge of the diagram, then it is visible (almost) all night, and hence in opposition. If the planet crosses the center horizontal line (the location of the Sun), then the planet is in conjunction with the Sun. If the trajectories of two planets cross in the diagram, then those planets are in conjunction with each other. If a number of planets are close together in the diagram, then they are all in conjunction with each other.

For example, midway through the year 2000, Mercury lags the Sun by about 2 hours, and Mars and Venus are in conjunction with the Sun. Jupiter and Saturn are close together during all of 2000, and are in opposition towards the end of 2000. Around May 2000 (near 2000.4 on the horizontal axis) Mercury and Saturn are all reasonably close together (in conjunction), but Uranus and Neptune do not participate. Midway through 2001, Mars, Uranus and Neptune are in opposition, Jupiter and Saturn are in conjunction with the Sun, and Venus is the morning star. The conjunctions of Mercury through Saturn around May 2002 and around September 2040 in the evening sky are also visible, and again Uranus and Neptune do not participate.

Fig. 7: Planets Diagram 2000-2001
Fig. 7: Planets Diagram 2000-2001
Fig. 8: Planets Diagram 2002-2003
Fig. 8: Planets Diagram 2002-2003
Fig. 9: Planets Diagram 2040-2041
Fig. 9: Planets Diagram 2040-2041




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