Mars' two tiny moons, Deimos and Phobos, circle the planet with blistering speed and never stay very far from it. When Phobos is at its greatest elongation from Mars, it is never more than three planetary radii from the planet. Since the moon circles Mars in just over seven hours, the window of opportunity to see it while it is away from the planet is only about an hour long. It is critically important therefore to know when the moon is at that point of greatest elongation and have a plan. Use an astronomy software program to plot Phobos position so you will know what it looks like relative to the planet. Make certain that you protect your night vision carefully and then execute your observing plan on time. At opposition, Phobos can range anywhere from about magnitude +10.2 when Mars is closest to Earth and down to magnitude +12 is Mars is near aphelion at the time of opposition. Once you have found Phobos, watch it for a little while to verify that you have not found a background star. The moon from greatest elongation should begin to move quickly back towards Mars. This motion should be discernible in as little as thirty minutes.
Deimos is not so subject to washout as Phobos is because it is not nearly so close to Mars at greatest elongation. Still it is just as important to find that time of greatest elongation to go looking for Deimos. Something else that is important that one does not normally worry about with solar system observing is a dark sky. Even with Mars at a perihelic opposition, Deimos will barely be brighter than magnitude +11.9. At less favorable oppositions, Deimos is beyond even the reach of medium size telescopes never becoming brighter than magnitude +13.7. I have viewed Phobos successfully under moderate light pollution with my 8-inch Cele-stron at favorable oppositions in 1988,2001 and 2003. Deimos is impossible to see under any light-pollution conditions because it is already near the limit of the 8-inch scope so a dark sky is absolutely critical to success. As with Phobos, verify that Deimos shifts its position over time. The motion is not nearly so rapid as that of Phobos but over the course of an hour the motion of Deimos relative to Mars should be apparent.
If the glare of Mars is a bit too much here is a dirty little trick you can try. Try running a ^-inch wide strip of aluminum foil across the field stop of your eyepiece (don't let the tape touch any optics!) and put the eyepiece in your telescope. The foil will form what is called an occulting bar. Use the bar to block Mars from your sight and this will eliminate the glare from the planet. Now see if Phobos, Deimos or both pop into view. You may have to experiment with the width of your occulting bar and it will have to vary in width depending upon magnification. Still, getting rid of that glare from Mars itself may prove to be the difference in seeing or not seeing the moons.
Observing Deimos and Phobos is a challenge that will stretch both your eyes, your sky and your equipment to their limits and may even require a bit of outside interference. They represent two of the great mysteries of the solar system in that no one can explain how a planet of relatively weak gravity managed to capture two asteroids into prograde circular orbits. For the amateur willing to do the work,they can be coaxed out into view. Do your studying and go find them.
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