The differential force is a kind of stretching and squeezing force that one encounters when considering large (not point mass) objects in a gravitational field.

     Since the force of gravity can be expressed as GMm/r² one runs into an interesting effect when one stops to consider heavenly bodies not as point masses, but as large bodies in space of irregular shape and composition.

     The first point to note here is that in such a body near to another massive object, some parts of the body are closer to the other massive object than others. Since the distance squared is a factor in determining the gravitational force acting between two masses, this results in parts of the "attracted" body being more or less attracted than others in such a way that the "attracted" body is actually stretched somewhat along the axis of attraction.

     This effect causes a distortion of the object's shape which varies periodically with the orbit of the "attractive" body resulting in the ocean tides which are so easily observed and in a roughly 11 centimeter land tide, which is not so apparent.

     The acceleration of a mass dm attracted toward a point mass M can be expressed by the following equation:

a=GM/r² (Eq. 1)

     For a roughly spherical object of radius R being attracted to a point mass M, the point nearest the mass M has an acceleration of

a₁ = GM/(r-R)²

     Whereas the point furthest from the mass M has an acceleration of

a₂ = GM/(r+R)²

     Given this one would find that the difference between the acceleration of one point and the other is given as follows:

=> Δa = GM(4rR/(r⁴-2r²R²+R⁴)) (Eq. 2)

     Given R<<r:

Δa =~ GM(4rR/r⁴) = GM(4R/r³) (Eq. 3)

     For the purposes of this discussion ?a can be referred to as the strength of the tidal, or differential force acting on a mass of radius R.

     This would lead one to understand that a free object in the presence of a gravitational field with a radius R>0 would be stretched. Figure 2 also describes a slight squeezing affect as well.

     Taking into consideration the extrema on the tangent plane of an orbiting spherical satellite one finds the acceleration to be:

a = GM(1/(r²+R²))

     Considering only the tangential component:

a = (R/v(r²+R²))GM(1/(r²+R²)) = RGM(1/(r²+R²)^3/2)

     Given R<<r:

a =~ RGM/r³ (Eq. 4)

     Thus it is found that the squeezing action of the differential force is proportional to the radius of the satellite.

     These stretching and squeezing actions are kept under control by the object's orbit and rotation and internal forces within a given object that hold it together similar to how a spring when pulled, resists change and pulls back towards its rest state. However, there is a point at which these stabilizing forces fail to compensate. This is point called the Roche limit and will be discussed later on.

     Some interesting consequences resulting from tidal forces include spaghettification, spring and neap tides, "tidal torque", orbital degradation, and the Roche limit.