Gamow-Manampiring

THEORETICAL MAXIMUM TIDAL ACCELERATION

Consider two planets of different masses orbiting a common center of mass (‘barycenter’) in an inertial reference frame. Assume both planets are rigid spheres with circular orbits. Therefore, to calculate the maximum tidal acceleration, we should start from this orbital mechanics formula:

T² / d³ = 4π² / [G (M1 + M2)] and

b = M2 d/(M1 + M2), where

T = orbital period

d = distance between the centers of the two planets

M1 = mass of the larger planet

M2 = mass of the smaller planet

b = distance from the “barycenter” to the center of the larger planet.

Suppose the above model is applied to the Earth-Moon system. We know that the barycenter of the system is within Earth. Now, consider a small object of mass m located on Earth's surface at its closest point to the Moon. The object of mass m naturally orbits with the Earth, so m experiences a centripetal force toward the barycenter,

F centripetal = m ω² (R - b), where

R = radius of the Earth

ω = angular velocity of the orbit = 2π/T

b = distance from the center of the Earth to the barycenter

Substituting the formulas for T and b into the centripetal force equation above, we obtain:

F centripetal = G m (M1 + M2) R/d³ - G m M2 /d².

The centripetal force is also the resultant of the real forces acting on an object of mass m. Consider the free-body diagram of object m:

w - N - F moon = G m (M1 + M2) R / d³ - G m M2 /d².

⇔ G m M1 /R² - N - G m M2 / (d - R)² = G m (M1 + M2) R/d³ - G m M2 /d².

⇔ N = G m M1 / R² - [G m M2 / (d - R)² - G m M2 / d²] - G m (M1 + M2) R / d³

It is clear from the normal force N ("apparent weight") that an object with mass m loses weight equal to the sum of the classical tidal forces and the Gamow-Manampiring forces.

Dividing both sides of the N above by m reveals the corrected theoretical maximum tidal acceleration, which is the sum of the classical tidal acceleration (in square brackets) and the Gamow-Manampiring acceleration:

a = [G M2 / (d - R)² - G M2 / d²] + G (M1 + M2) R / d³

Using the same principle, we can derive the maximum tidal acceleration for a position on the side of the Earth farthest from the moon. Consider an object with mass m there:

F centripetal = m ω² (b + R),

⇔ w + F month - N = G m (M1 + M2) R / d³ + G m M2 / d²

⇔ G m M1 / R² + G m M2 / (d + R)² - N = G m (M1 + M2) R / d³ + G m M2 / d²

⇔ N = G m M1 / R² + G m M2 / (d + R)² - G m M2 / d² - G m (M1 + M2) R / d³

⇔ N = G m M1 / R² - [G m M2 / d² - G m M2 / (d + R)²] - G m (M1 + M2) R / d³

Divide both sides of N by m, then clearly a part that showing the theoretical maximum tidal acceleration,

a = [G M2 / d² - G M2 / (d + R)²] + G (M1+M2) R/d³

Bibliography:

Thomas Arny, Explorations: Introduction to Astronomy (McGraw-Hill, 2003).

Rebecca Boyle, Our Moon: A Human History (Scepter, 2024).

Helen Czerski, Blue Machine: How the Ocean Shapes Our World (Torva, 2023).

David George Bowers & Emyr Martin Roberts, Tides (Oxford University Press, 2019).

George Gamow, Gravity (Anchor Books, 1962).

Paul G. Hewitt, Conceptual Physics (Little, Brown and Company, 1985).

I.M. Longman, “Formulas for Calculating Earth's Tidal Acceleration Due to the Moon & Sun” (Journal of Geophysical Research, December 1959).

William Lowrie, Geophysics: A Very Short Introduction (Oxford University Press, 2018).

Jos Manampiring, “What Causes Ocean Tides?” (Facebook).

Jos Manampiring, “The Tendency of the Ocean to Break Away from the Earth” (Facebook).

Jos Manampiring, “The Emergence of the Gamow-Manampiring Perspective” (Facebook).

Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy (University of California Press, 1999).

Hugh D. Young & Roger A. Freedman, Sears & Zemansky's University Physics (Addison-Wesley Publishing Company, 2000).

Oxford Dictionary of Physics (Oxford University Press, 2015).

https://science.nasa.gov/moon/tides/

Author: Jos Manampiring