# Primer on Mars oppositions(by Jacques Laskar, IMCCE, Paris Observatory)

(version française)

### Mars opposition

The opposition of Mars is the moment when the geocentric longitude of Mars (counted on the ecliptic of the date) is contrary to the geocentric longitude of the Sun.

If the orbits of two planets were circular, this moment would be also a minimum of distance between two planets. As the orbits slightly eccentric and are inclined, the date of the minimum of distance can differ from it from a few days.

Example ( IMCCE data) :
On 27/08/2003 at 09h 51m 14s UT : the Earth-Mars distance is minimal.
On 28/08/2003 at 17h 58m 49s UT : Mars is in opposition.

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### Duration between two oppositions:

The mean angular (sideral) velocity of the Earth is n1 = 6.28307585 radians/yr, and for Mars n2 = 3.34061243 radians/yr. If we start at an opposition, the next one occurs when the Earth catch up with Mars, making one additional lap over the Sun, that is
n1*t = n2*t + 2 pi
thus, after (in years)
t1 = 2 pi/(n1-n2)
which can be converted in number of revolutions (so practically in years)
nopp = t1/T1=n1/(n1-n2) = 2.135311456
The next opposition will thus occur in a little bit more than 2 years and 49 days.

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### Earth-Mars distance at opposition

If the orbit of the Earth and Mars were circular and coplanar, the Earth-Mars distance at opposition would be constant and equal to the difference of their semi-major axis, that is dmin ~ 1.524 - 1.000 = 0.524 au (astronomical unit ~150 million of km (Mkm)).

As it is not the case, this distance to the oppositions will vary from one opposition to another. As the eccentricity of Mars is dominant, the minimal distance will be obtained in the vicinity of the perihelion of Mars (minimal Sun-Mars distance).

Fig.1: Configurations of the oppositions of Mars between 1980 and 2003

Minimal Earth-Mars distances in the vicinity of the recent oppositions.

```
date             Earth-Mars distance      Mars apparent
(au)        (Mkm)        diameter

26 february  1980      0.677310    101.324         13.83"
5 april     1982      0.635114     95.012         14.75"
19 may       1984      0.531461     79.505         17.63"
16 july      1986      0.403568     60.373         23.21"
22 september 1988      0.393147     58.814         23.83"
20 november  1990      0.516922     77.330         18.12"
3 january   1993      0.626093     93.662         14.96"
11 february  1995      0.675694    101.082         13.86"
20 march     1997      0.659382     98.642         14.21"
1 may       1999      0.578463     86.537         16.19"
21 june      2001      0.450167     67.344         20.81"
27 august    2003      0.372719     55.758         25.13"
30 october   2005      0.464063     69.423         20.19"
```

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### Favorable oppositions

From a favorable opposition, i.e. an opposition taking place at least distance
between the orbit of the Earth and Mars (as it will be about the case on August 27, 2003),
one will return to the same configuration when, after a whole number N of oppositions,
the Earth will have carried out a whole number of turns, that is to say when

N *nopp is close to an integer value.

We thus search for the rational approximations M/N of nopp, which is easily done with the continuous fraction expansion of nopp

nopp ~ 2+1/(7+1/(2+1/(1+1/(1+1/(3+1/(1))))));

which successive rational approximants are

```
aa1 = 2 + 1/7    = 15/7    = 2.14285714285714
aa2 = 2 + 2/15   = 32/15   = 2.13333333333333
aa3 = 2 + 3/22   = 47/22   = 2.13636363636364
aa4 = 2 + 5/37   = 79/37   = 2.13513513513514
aa5 = 2 + 18/133 = 284/133 = 2.13533834586466
aa6 = 2 + 23/170 = 363/170 = 2.13529411764706
```
There will be thus a favorable opposition every approximately 15 years, after 7 oppositions, but one will approach the starting configuration still better every 32 years, 47 years, 79 years, 284 years, or 363 years. Of course these calculations are approximate, because they do not take account of the deformation of the orbits. .

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### Earth-Mars distance at favorable oppositions.

The favorable oppositions take place every approximately 15 years, but the minimal geometrical distance between the two orbits is reached with this favorable opposition only when the two planets are very close to this optimal position at the time of the opposition. Fig.2 : Minimal Earth-Mars distance with the most favorable oppositions, of -300
at + 300 years from January 1, 2000 to 12 H (J2000) (red circles).
The green curve is the minimal distance between the orbit of the Earth and Mars
(thus the minimal distance enters the two ellipses).

### Earth-Mars distance evolution over 1 million years.

Because of the perturbations of other planets, the orbits of the Earth and Mars become deformed slowly in time and precesse slowly in their plane and in space. The configuration of the two orbits thus evolves, and the minimal Earth-Mars distance too. Practically, the minimum of distance will take place when the eccentricity of Mars is maximum. As we saw above, the minimal distance from the two orbits is not always carried out with the oppositions, but one can consider that this minimum is roughly reached over 100 years (see figure 2). Over a long duration, one can thus be satisfied to calculate the minimal distances between the two orbits, without explicitly seeking the dates of the favorable oppositions which will achieve (almost) these minima (fig. 3).

Fig.3: : Minimal distance (D_TM) between the two orbits of the Earth and Mars
from -500 000 to + 500 000 years since J2000, in million km
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### Remarkable dates

On a few thousands of years around the current date, the minimal Earth-Mars distance decreases (fig. 3). It was thus larger in the past, and will be smaller in the future. But this minimal distance is carried out only very approximativemnt during the oppositions. The principal characteristic of the opposition of August 2003 is to achieve this minimal distance with a very good approximation, which occurs only all the 79 or 284 years. In the past, there was a very good realization of this minimum 79 years ago, in August 1924, but as the minimal distance between the orbits of the Earth and Mars was larger, this minimum is larger than the minimum of August 27, 2003. To find an equivalent Eart-Mars distance, it is necessary to go much further in the past, when the eccentricity of Mars was larger, 59 618 years ago (the 73 000 years value often quoted corresponds to a crude approximation).

In the next millenium, as the minimal distance between the orbits of the Earth and Mars decrease, one will find many oppositions with a Earth-Mars distance closer than that of August 27, but the first will take place only in 284 years, dates to which this minimum will be almost achieved at the time of a favorable opposition (Fig.2).

```Minimal distances of the oppositions from -300 to + 300 with respect to J2000 Only minimal distances lower than 56 Mkm are given (computations from ASD/IMCCE, Observatoire of Paris)

date               Earth-Mars distance        apparent Mars
(ua)           (Mkm)        diameter

25 august 1719       0.374010       55.951          25.05"
13 august 1766       0.373260       55.839          25.10"
18 august 1845       0.373021       55.803          25.11"
22 august 1924       0.372846       55.777          25.12"
27 august 2003       0.372719       55.758          25.13"
15 august 2050       0.374051       55.957          25.04"
30 august 2082       0.373564       55.884          25.08"
19 august 2129       0.373276       55.841          25.10"
24 august 2208       0.372794       55.769          25.13"
28 august 2287       0.372254       55.688          25.16"

```

The numerical integrations carried out over 500 000 years (ASD/IMCCE) also make it possible to specify the following dates:

Last opposition where the Earth-Mars distance was smaller than that of August 27, 2003:

59 615.267 years before J2000, d_TM = 0.372452 au (55.718 Mkm), Mars apparent diameter : 25.15".

On the other hand, while looking at even further, one will find values of the Earth-Mars distance even smaller, with a minimum in the last 500 000 years in

80 955.939 years before J2000,d_TM = 0.358739 au (53.667 Mkm), Mars apparent diameter : 26.11".

Next the grand minimum will take place (Fig.3)

23 695.342years after J2000, d_TM = 0.359769 au (53.821 Mkm), Mars apparent diameter : 26.04".

and the absolute minimum in next the 500 000 years will take place

292 851.722 years after J2000, d_TM = 0.358541 au (53.637 Mkm), Mars apparent diameter : 26.13".

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© J. Laskar, IMCCE/CNRS, Observatoire de Paris, (last revision 14 août 2003)