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Solar System Gravitation
Name: Kevin
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: N/A
Question:
It is my understanding that the orbits of the
planets are influenced mainly by the mass of the sun, with some
lesser influences from other orbiting bodies. How is it possible
for the sun to exert a gravitational pull on bodies so far away to
keep them in orbit, especially with the high velocities these
bodies orbit at? As massive as the sun is, I cannot imagine how it
would hold the outer planets in its grasp, being so minuscule a
body in respect to the distances involved.
Replies:
The Sun is hardly "minuscule," as it comprises practically all of the
mass of the Solar System. The force of gravity decreases as the square
of the distance from the attracting object (1/d^2), which never actually
goes to zero, no matter how large the distance d becomes.
The Solar System also orbits the center of the Milky Way galaxy,
although that is much farther away than the Sun is from the planets.
Our galaxy, in turn, is being pulled toward the Virgo cluster of
galaxies, which is farther still...
Richard Barrans
Department of Physics and Astronomy
University of Wyoming
Great question. It’s true that even distant planets have fairly high orbital
(“tangential”) velocities, although the velocities do decrease with distance
from the sun. But what’s more critical is the *angular* velocity. For example,
Pluto (ignoring its recent demotion from “planetary” status) is tangentially
whizzing through space fairly quickly, but because it takes nearly 250 earth
years to orbit the sun, its angular velocity is not high, and that’s what determines
centrifugal force.
To see this mathematically, recall centripetal force is m*v^2 (where v is tangential
velocity) and gravitational force is g*m1*m2/r^2. Setting these forces equal to each
other, you arrive at G*m = r*v^2 (m here is the sun’s mass). G and m are constants, so
this says as r (planets distance to sun) multiplied by *tangential* (not angular)
velocity equals a constant. If you make a table of the planetary distances and their
corresponding *tangential* velocities squared, and multiply them, you’ll see a pattern
emerge.
On a related topic, some people (my younger self) have been puzzled that both Jupiter and
Pluto could both be outer planets and have stable orbits with such a mass difference (when
younger, I thought Jupiter should be "flung off"). You have probably realized that planetary
*mass* plays no role. Massive planets have stronger gravitational force, but that’s balanced
by an equally stronger centrifugal force, and vice versa.
P. Bridges
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Update: June 2012
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