Sunday, December 30, 2012

A lesson in astronomy and geometry

John Goss

John Goss is chairman of the Mid-East Region of the Astronomical League and a former president of the Roanoke Valley Astronomical Society.

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While our universe is incredibly vast, it isn't so complex that we can't unravel its secrets and discover how it works -- but only if we look carefully enough at what lies around us. The night of Jan. 21 presents us an opportunity to measure the distance to our closest neighbor in space, the moon, and to estimate its approximate diameter. However, you will need help from your friends in South America.

About 11 o'clock that night, take a good look at the bright waxing gibbous moon. About one apparent lunar diameter to its north shines Jupiter, our solar system's largest planet. It appears as a pinpoint because, even though Jupiter's diameter is 40 times greater than the moon's, it lies more than 1,500 times farther away.

It is time to call your friend living near Concepcion, Chile. Be very polite. He may be a little perturbed because it is midnight down there. He may become more upset when you ask him to take a quick look at the moon to see if there is anything unusual about it. If he still is speaking amicably to you, he'll most likely report that there is a strange object hugging its south rim, almost obscured by the lunar glare. Quickly explain that the object is Jupiter and that, from your viewpoint in Roanoke, it lies above the moon, not below it. Then politely hang up before he gets mad.

Next, call your friend in Cali, Colombia, just south of Bogota. Don't worry, as it is the same time there as it is here. Ask him to look at the moon. As with your Chilean friend, he likely will say that there is an object almost touching its northern rim. From his point of view, Jupiter has moved to the other side of the lunar disk.

Now, you almost have all the information you need to figure out the moon's diameter and distance.

Because Jupiter is very far away, its light rays travel in a parallel manner toward Earth. This means that, as viewed from Earth, the moon's diameter equals the distance on Earth between where Jupiter appears on opposite edges of the moon's disk. In other words, the lunar diameter is the same as the straight line distance from Cali, Colombia, to Concepcion, Chile -- or about 2,300 miles. (Earth's curvature adds 300 miles to the straight line distance.) The moon's true diameter is just under 2,200 miles.

Finding the moon's distance is slightly more complicated. You first need to know that the moon's angular width in the sky covers one-half degree.

-- Trace a full 360-degree sky circle starting at the moon, passing through Jupiter, dropping to the northern horizon, traveling below your feet, and swinging back up to the moon. Because the moon spans only one-half degree, the giant circle's circumference equals 720 moon diameters.

-- Remember that the straight-line distance from Cali to Concepcion is almost 2,300 miles.

-- If a circle with a radius equaling the Earth-moon distance were centered on the moon, the 2,300 miles between Cali and Concepcion would equal one-half degree, the moon's angular diameter. Because a complete circle equals 360 degrees, the circumference of the Earth-moon circle would be 2,300 miles times 720, which equals 1,656,000 miles.

-- Remembering your elementary geometry, the circumference of a circle equals twice its radius multiplied by pi (which equals about 3.14). The distance to the moon, i.e., the radius of the giant circle, is calculated to be 260,000 miles. The accepted value for the Earth-moon distance on Jan.21 is 249,000 miles.

Not a bad start for understanding our universe!

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