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Vis-Viva along with the Geminid Meteoroid Stream


Annually in December, our planet Earth travels through the orbital path of a mysterious solar system object called 3200 Pheathon. Debris and rubble trailing behind Phaethon apparently fills this asteroid's orbital path nearly full circle. For every year, around December 13/14, bits and pieces from Phaethon burn up in Earth's atmosphere as the annual Geminid meteor shower, adorning the nighttime with one of the best meteor diplays of the year. For more on observing the December Geminid meteor shower, check out Meteor Showers Online.

Meteor Speeds

The Geminid meteors are said to hit the Earth's atmosphere at a moderate 35 kilometers (22 miles) per second. At the higher end of the scale, the November Leonids fly at 71 kilometers (44 miles) per second; and at the lower end of the scale, the October Draconids travel at a languid 20 kilometers (14 miles) per second. For the velocities at which other shower meteors crash into our atmosphere, click here.

Velocity of the Geminid Meteoroid Stream

Before these bits and pieces from Phaethon vaporize as meteors in our atmosphere, they're called meteoroids when streaming about in outer space. If we wish to find out how fast Phaethon or these meteoroids travel when at the Earth's distance from the Sun, all we need to know is the semi-major axis of its orbit in astronomical units. An astronomical unit represents the distance between the Earth and the Sun. This site gives the semi-major axis for Phaethon's orbit as 1.271 AU (astronomical units).

The Vis-Viva Equation

In the Vis-viva equation listed below, r = distance from the Sun in astronomical units; and a = the semi-major axis of the orbiting body in astronomical units. Since we're figuring the velocity of the meteoroids at the Earth's distance from the Sun, r = 1 astronomical unit and a = 1.271 astronomical units. (The 66,627 figure represents the Earth's orbital velocity in miles per hour. If preferable, you may replace this figure with 107,225, the Earth's orbital velocity in kilometers per hour.)
Here is the Vis viva equation (r = 1; a = 1.271):
velocity = 66,627 x the square root of (2/r - 1/a)
velocity = 66,627 x the square root of (2/1 - 1/1.271)
velocity = 66,627 x the square root of (2 - 0.78678)
velocity = 66,627 x the square root of 1.21321
velocity = 66,627 x 1.10146 = 73,387 miles per hour
At the Earth's distance from the Sun, these meteoroids travel at 73,387 miles per hour, 1,223 miles per minute or 20.4 miles per second. This is only marginally slower than the 35 kilometers (22 miles) per second that the Geminid meteors strike Earth's atmosphere. This leads me to think that where the orbits of Earth and the Phaethon meteroid stream meet, Earth isn't really going toward or away from the meteoroid stream. In the case of the November Leonid shower, however, there is a wide divergence between the meteoroid and the meteor speeds. For a short write-up on the November Leonids, click here.

Perihelion and Aphelion Velocities of the Geminid Meteoroid Stream

Phaethon's orbit around the Sun is enormously eccentric. When Phaethon or its meteoroid stream swings closest to the Sun (perihelion), it is only 0.14 astronomical unit away from the Sun. When farthest from the sun (aphelion), the orbit looms 2.401 astronomical units distant. With this information, we can figure how fast Phaethon or its meteroid stream travels at perihelion or aphelion. (By the way, to see a super cool animation of the orbits of Phaethon and the planets, click here.)
At perihelion, r = distance from the Sun = 0.14 astronomical units; and a = semi-major axis = 1.271 astronomical units:
velocity = 66,627 x the square root of (2/r - 1/a)
velocity = 66,627 x the square root of (2/0.14 - 1/1.271)
velocity = 66,627 x the square root of (14.2857 - 0.7868)
velocity = 66,627 x the square root of 13.4989
velocity = 66,627 x 3.6741 = 244,794 miles per hour at perihelion
At aphelion, r = 2.403 astronomical units; and a = 1.271 astronomical units:
velocity = 66,627 x the square root of (2/2.403 - 1/1.271)
velocity = 66,627 x the square root of (0.8323 - 0.7868)
velocity = 66,627 x the square root of 0.0455
velocity = 66,627 x 0.2133 = 14,214 miles per hour at aphelion

copyright 2007 by Bruce McClure

Resources:
Curious About Astronomy? Ask An Astronomer
Vis-viva Equation by Wikipedia
Rocket Science ain't all Rocket Science by Leslie S. Coleman (Frosty Drew Observatory)
Velocity comes from vis viva equation


November 2007 Feature * January 2008 Feature