FG = G ---- nt
where G is the gravitational constant, 6.67 x 10-11 nt⋅m2/kg2, m is the mass of the other celestial body, mE is the mass of the earth, 5.98 x 1024 kg, and d is the closest distance between the earth and the other body.
Gravitational Field Strength: Gravitational field strength is the gravitational force (due to another planet in the solar system) divided by the Earth's own mass. P = FG / mE, measured in Newtons per kilogram.
P = G --- N/kg
N.B. The gravitational force between two objects is a vector, and the gravitational force on many planets (and sun and moon) on the earth is a vector field, with a net direction and magnitude. Therefore, the gravitational field strength is also a vector field. However, the quantity P used here is just the magnitude of the field.
We'll use this quantity to compare the effects of the relative amount of influence of the Earth's motion by the bodies in our solar system.
Orbital Closest Field Strength
Mass Radius Distance acting on Earth
Body (kg) (m) (m) (N/kg) (ratio)
Sun 1.97E+30 1.50E+11 1.50E+11 5.81E-03 100
Moon 7.35E+22 3.84E+08 3.84E+08 3.30E-05 0.6
Jupiter 1.90E+27 7.80E+11 6.30E+11 3.18E-07 0.0055
Venus 4.87E+24 1.08E+11 4.20E+10 1.83E-07 0.0032
Saturn 5.69E+26 1.43E+12 1.28E+12 2.30E-08 0.0004
Mars 6.42E+23 2.28E+11 7.80E+10 7.01E-09 0.00012
Mercury 3.31E+23 5.85E+10 9.15E+10 2.62E-09 0.00005
Neptune 1.03E+26 4.51E+12 4.36E+12 3.60E-10 0.00001
Uranus 8.71E+25 2.88E+12 2.73E+12 7.76E-10 0.00001
All the planets are in a roughly circular orbit around the sun. Since the orbits have different radii, each planet has a shortest distance to the earth with a frequency that varies with difference in radii. The "closest distance" column is the difference in radii (Δr = |rEARTH - rOTHER| - that's the closest distance any planet can come to the earth.
The last column is gravitation field strength normalized to 100. The sun's effect was arbitrarily scaled to 100, and all other potential values are shown scaled to that number to give you an idea of how small the effects are of the perturbations on our planet.
The results are what you expect: The sun has the most influence by a far amount. The other planets perturb our orbit only weakly. The moon has the largest effect on the earth. It is so heavy relative to the earth that the moon doesn't actually revolve around the earth. Instead the earth and the moon revolve around a center-of-mass point located about 1000 km below the surface of the earth. A good illustration of that motion can be found here.
The sun is 390 times further away than the moon, but it has 27 million times more mass. Its influence on the earth is over 100 times stronger.
The sun has over a thousand times more mass than the next closest planet, the gas giant Jupiter. Because of this, the solar system can be thought of as having one gigantic central mass, whose little planets affect its orbit hardly at all.
The planets have even less of an effect on the earth's motion - miniscule compared to the nearby moon. Jupiter and Venus have about equal effect, but the gravitational field strength that they exert on the earth is 100-200 times weaker than the moon's. Other planets have even weaker influences.
Sincerest thanks to Oolong for pointing out problems with my terminology. There's an important difference between gravitational field strength and potential.