[Chaos-l] How big Mars gets, et al.

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Hi - I had fun with my calculator writing this article. Hope you enjoy.
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Mars as Big as the Moon?

By Mike Gallagher, Chapel Hill Astronomical and Observational Society (CHAOS)

There was lots of internet activity in August touting a rare, but imaginary, appearance of Mars where it would look as big as the moon. Mars does get wonderfully bright this fall as our Earth scoots by the red planet in a 26 month orbital cycle. However, even at its closest, Mars doesn?t get to look more than about 1/60th as big as the moon. But while we are thinking about heavenly objects which take up a fair piece of the sky, there are some great choices. For a casual observer there are beautiful star clusters, glowing interstellar clouds, and galaxies which do look as big in our sky as the moon. It can be a fun fall outdoor activity to spot some faraway objects in the sky. Plus, it is a nice math indoor exercise to grab a calculator and figure out how big things look. The math ties in with objectives students are meeting now that school is back in session -- find ratios and proportions in the state math curriculum!

It?s common sense that an object looks bigger when it?s closer. When we buy the cheapest ticket to a ball game, it?s no surprise that the players look tiny. Angular measure provides a convenient number for how big something appears. If a movie screen is all around us, that would be 360 degrees. If, from our seat at the game, the quarterback looks about as big as the width of an outstretched pinky, that?s about 1 degree. The moon and the sun each appear to be about half a degree. 

For small angles, say, less than 10 degrees, there is a pretty simple way to estimate how big an object looks (angle measure) based upon how big an object is (length measure), and its distance from an observer. First find the ratio of length to distance. Next multiply the length to distance ratio by 180 and divide by pi to get the angle in degrees. For example, the moon is 2140 miles in diameter and is about 222,000 miles away. The length to distance ratio is 0.00966 (about 0.01). Multiplying by 180/pi gives us 0.552, slightly over half a degree. Since each degree is divided into 60 minutes, we can get multiply by 60 and get 33 arc minutes for the moon. Another way to look it is that if you divide an object?s length by its distance and get about 1 hundredth (0.01), the object looks about the same size as the full moon.

Mars is 4,222 miles in diameter and an unusually close orbital pass would put Mars at 35 million miles from Earth. The angle calculation for Mars dividing 4,222 by 35 million gives a length to distance ratio of 0.000121. Multiplying by 180/pi to get degrees and then by 60 to get minutes gives 0.4, about 1/2 of an arc minute, way tinier than the Moon! Of course, going back to the email flurry, maybe someone just saw the ? (minute) for Mars and ? (degree) for the moon and figured that meant they were the same size!

However, there are certainly objects further away than Mars that appear at least as large as the Moon. Do the math for the sun. The sun?s diameter is about 865,000 miles and its distance is 93,000,000 miles. The sun?s ratio of length to distance would be 0.009. So the sun comes out to be just over half a degree or about 32 minutes, very close to the apparent size of the moon. 

What else is at least as big as the moon? Here are several objects which can be seen with the unaided eye at a dark sky location or with binoculars where most of us live. 

In our own spiral arm of the Milky Way Galaxy is a beautiful star cluster called the Pleiades. At a size of 2 degrees, the Pleiades (M45) cluster is high in the morning sky in September, just East of where Mars is in 2005. There are about 100 youngster stars (about 70 million years old) born from the same cloud in the cluster. Since the cluster is quite remote at 407 light years, it needs to be rather wide (14 light years) to look so big, giving a length to distance ratio of 0.034.

Leaving our spiral arm, the Lagoon Nebula (M8), at a distance of 5,200 light years, is about 10 times as far away as the Pleiades, but the Lagoon is still quite big in our sky, at about 90 minutes by 40 minutes. Look low in the Southern sky in September toward the center of our galaxy in Sagittarius. The Lagoon is a younger star forming cloud, where the new, highly energetic stars are causing the Hydrogen gas to light up in pinks and reds. As a bonus, the Trifid Nebula (M20) is a nearby relative.

Now jump out of the milky way disk of stars to an ancient, spherical cluster, which orbits the center of the galaxy. The Hercules Cluster (M13) includes about half a million stars in a tight formation. M13 is 23,400 light years away, so it is almost 10 times as far as M8, but you can estimate its diameter, since it takes up about 1/3 of a degree in the sky, almost as big as the moon!

Turn toward the northwest (about 10 to11 p.m. in September) to leave our galaxy entirely and view the nearest spiral, the Andromeda Galaxy (M31).  Andromeda is 2.3 million light years away and takes up about 3 degrees by 1 degree in the sky. Being so far away, it isn?t surprising that Andromeda is rather long. The estimate is 130,000 light years for this galaxy of about 200 billion stars. The Andromeda Galaxy looks very pretty with binoculars, but it might take staying up late or getting up early in September to have a look. 

Objects over quite a large span in the universe have vastly different sizes and distances, but some of them share a pretty close ratio of length to distance and therefore conveniently fit in similar niches in the sky.