Can Mars and the Moon have the same size?

From a recent e-mail




In newspaper articles and in e-mails on the Internet there was a buzz of excitement:
Keep the night of August 27, 2003 free, and look up at the sky for a spectacular view of Mars.

And indeed, when you were lucky enough to have a clear sky, Mars could be seen as a very bright reddish star.
But not more than that. And still about 75 times smaller than the moon!

So, what has been happening?

Planetary orbits

All planets are orbiting the sun, due to the force of gravity.
Their motion is described by the three Kepler laws.

  1. The planetary orbits are circles or ellipses
  2. The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse.
  3. The ratio between the square of a planet's period and the third power of the planet's distance to the sun is a constant.

The third law tells us that the period will be longer for a planet which has a larger distance to the sun.
For example:

  • Earth: average distance 150 million km, period 365 days
  • Mars: average distance 228 million km, period 687 days

Average distance, because the orbits are elliptical.

The earth orbit is only slightly elliptical. When the earth is closest to the sun (this point is called the Perihelion), the distance is 147 million km, whereas in the Aphelion (when the distance is largest), this distance is 152 million km.

The Mars orbit is more elliptical: The perihelion distance is 207 million km and the aphelion distance 249 million km.

The sketch tries to give an impression of the orbits of the inner planets. The orbit of Mercury is quite elliptical, whereas the Venus orbit is almost perfectly circular. Even for the earth is not easy to see in the sketch that the orbit is not a circle. For Mars it is clearer, the perihelion is situated in the lower right part of the sketch

Opposition and Synodic Period

As the Earth is making its rounds faster than Mars, the Earth will continuously overtake Mars. The overtaking will occur when Mars and Earth are aligned with, and at the same side, of the Sun. This overtaking is called an Opposition.
In the sketch above Mars and Earth are almost in opposition.

How much time between Oppositions? The easiest way to calculate this goes as follows:
In a time of 365*687 days, the Earth will make 687 rounds, and Mars will make only 365 rounds.
This means that there will be (687-365)=322 overtakings in that time.
This gives for the time between oppositions: 365*687/322=779 days, about 26 months.
This time needed for one 'overtaking' is called the Synodic Period.

The distance between Mars and Earth

The distance between Mars and the Earth will vary a lot, depending upon the positions of both planets. During opposition this distance is minimal, after which it will increase for more than a year. As a result Mars will become less and less bright and even become invisible for many months because it will be too near and even 'behind' the Sun. Then Mars will appear again and become brighter and brighter until the next opposition, etc.See the picture below for a sketch of the apparent size of Mars before, during and after the present opposition.

If both orbits were circular, this would be the end of the story.
During opposition the distance would be always (228-150)=78 million km.

However, the orbits are elliptical and that makes the story more complicated but also more interesting.

The abolute minimal distance between Mars and the Earth would occur when during an opposition Mars would be in its perihelion and at the same time the Earth in its aphelion! This would result in a distance of (207-152)=55 million km!

On the other hand, when during opposition Mars is in its aphelion and at the same time the Earth in its perihelion, the distance would be almost double: (249-147)=102 million km.

In the picture to the right a sketch is given of the oppositions between 1995 and 2010.
Because the synodic period is (779-687)=92 days longer than the Mars period, the oppositions move forward along the Mars orbit.

At the 1995 opposition Mars was almost in its aphelion, the distance was still 101 million km. The following oppositions (1997-1999-2001) became brighter and now, in August 2003, Mars is almost in its perihelion, resulting in a distance of 55.6 million km, only marginally longer than the absolute minimal distance! The next oppositions will give again longer distances. From the figure it can be seen that this whole cycle repeats with a priod of about 16 year.

Roughly every 16 years the opposition will result in a short distance. For example:

  • 1971: 56.2 million km
  • 1988: 58.7 million km

It will be clear now that every 16~ years the distance at opposition will be small, 55-60 million km. During such an opposition Mars will be very bright, comparable to Venus and Jupiter.
But NOT like the moon.
The apparent size of the moon is half a degree or 1800 arcsecs. The size during an perihelic opposition is about 24 arcsecs, so even under these favorable conditions you will need a power 75x telescope to see Mars like the moon!

Is there then nothing special about the August 2003 opposition?

Well, as mentioned above, the distance during this opposition was almost as small as it can be. Computer calculations have shown that this distance has not been so small during many tens of thousands of years.
So in that sense this opposition is special, although the effect is not observable, as the difference in distance with for example the 1971 oppostion is only 1 %.

The main reason is that the orbits of the planets are not completely stable.
The alignment of the ellipses (and other properties) changes on time scales of many thousands of years.
At present the orbits of Mars and the Earth are more or less aligned with the perihelion of Mars in about the same direction as the aphelion of the Earth.
So in the next hundreds of years more of these close oppositions will occur.