Download Vignette 2 Gravitational Perturbations and the Prediction of New

Vignette 2
Gravitational Perturbations and the Prediction of New Planets
Computing the orbit of the earth as an ellipse around the center of mass for the earth-sun system
assumes that they are the only two masses in the universe. In reality, the Universal Law of Gravitation
implies that the earth interacts gravitationally not only with the sun, but with every other mass in the
Universe: the moon, the other planets, asteroids and comets, the distant stars.
The Two-Body Approximation
However, from the form of the gravitational force, we see that the interactions are largest when two
situations are fulfilled: 1) the product of the masses of the two objects is large, which maximizes the
numerator of the expression for
the strength of the gravitational
force, and 2) the objects are near
each other, which minimizes the
denominator of the force equation.
The two-body approximation
that the orbits of the planets
are determined only by the
gravitational interaction between
the sun and the planet is possible
because:
The sun is so massive
compared with every other
object in the solar system.
Objects outside the solar
system, such as stars, are
so distant that the distance
squared factor in the
denominator renders their
gravitational interactions
with the planets negligible. For example, the sun is about 300,000 times more massive than the earth,
and about 1000 times more massive than the largest planet. Thus, the product of the mass of a planet
and the mass of the sun is always much larger than the product of the masses of any two planets, and it
is a good initial approximation to neglect all interactions except that of the planet and the sun.
Gravitational Perturbations
However, the small deviations from this ideal picture have consequences if careful measurements are
made. These small deviations from the simplified picture are called perturbations. They can be
calculated systematically using Newton’s laws of motion and gravitation from the positions of the
known masses in the solar system.
If we account carefully for all known gravitational perturbations on the motion of observed planets, and
the motion of the planet still deviates from the prediction, there are two options:
 Newton’s Law of Gravitation requires modification.
 There is a previously undetected mass that is perturbing the orbits of the observed planets.
Exercise
1. What happens to the gravitational force if the distance between objects is increased?
2. The interactions between two bodies is greatest when what two situations are fulfilled?
3. Compare the products of the masses of the Sun and Mars with the product of the masses of
Mars and the earth.