M as: Pulling out the R2 term from the denominator gives: The Maclaurin series of is which gives a series expansion of: The first term is the gravitational acceleration due to M at the center of the reference body , i.e., at the point where is zero. This term does not affect the observed acceleration of particles on the surface of m because with respect to M, m (and everything on its surface) is in free fall. When the force on the far particle is subtracted from the force on the near particle, this first term cancels, as do all other even-order terms. The remaining (residual) terms represent the difference mentioned above and are tidal force (acceleration) terms. When ∆r is small compared to R, the terms after the first residual term are very small and can be neglected, giving the approximate tidal acceleration for the distances ∆r considered, along the axis joining the centers of m and M: When calculated in this way for the case where ∆r is a distance along the axis joining the centers of m and M, is directed outwards from to the center of m (where ∆r is zero). Tidal accelerations can also be calculated away from the axis connecting the bodies m and M, requiring a vector calculation. In the plane perpendicular to that axis, the tidal acceleration is directed inwards (towards the center where ∆r is zero), and its magnitude is in linear approximation as in Figure 4. The tidal accelerations at the surfaces of planets in the Solar System are generally very small. For example, the lunar tidal acceleration at the Earth's surface along the Moon–Earth axis is about , while the solar tidal acceleration at the Earth's surface along the Sun–Earth axis is about , where g is the gravitational acceleration at the Earth's surface. Hence the tide-raising force (acceleration) due to the Sun is about 45% of that due to the Moon. The solar tidal acceleration at the Earth's surface was first given by Newton in the Principia. See also Amphidromic point Disrupted planet Galactic tide Tidal resonance Tidal stripping Tidal tensor Spacetime curvature References External links Gravitational Tides by J. Christopher Mihos of Case Western Reserve University Audio: Cain/Gay – Astronomy Cast Tidal Forces – July 2007. Myths about Gravity and Tides by Mikolaj Sawicki of John A. Logan College and the University of Colorado. Tidal Misconceptions by