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Michael Koohafkan What is Tidal Locking? Masses exert gravitational pull on other objects The Moon and Earth pull each other as the moon orbits the Earth Gravitational pull of Moon slows rotational speed of Earth Tidal locking occurs when Earth’s rotation becomes synchronized with Moon’s orbit. What would tidal locking look like for us? Days would be as long as the orbital period of the Moon. The Moon would appear to be ‘locked’ in place in the sky, and would only ever be visible on one side of the world. Does tidal locking only occur between a mass and a satellite? No. Tidal locking can occur between any two masses that orbit around each other. Planets can become tidally locked with the stars they orbit around, and stars in a binary system can become tidally locked together. How can tidal locking be quantified? Gravitational pull between masses can be measured Rotational and orbital speed of masses can be measured Rate of change of rotational speed can be calculated. If it can be represented as a function, then approximate length of time until tidal locking can be calculated Why does any of this matter anyway? Tidal locking can help measure the age of a planet in relation to a satellite. By measuring the rate at which a planet or satellite is approaching a tidal lock, we can extrapolate back and estimate the age of a satellite or planet. Tidal locking can have interesting environmental effects that we can observe on other planets. The Math Tidal Locking can be represented simply (and primitively) with a linear function Further accuracy requires the use of a few relationships Important Factors Sun’s gravitational pull has effect on Earth-Moon system Moon’s gravitational pull is responsible for tidal locking Moon’s rotational speed *may* be changing Moon’s orbital speed *may* be changing Moon is drifting away from Earth Moon’s gravitational pull is decreasing due to increase in distance The Equations Depending on what factors we want to ignore, we can quantize this relationship in a simple or complex manner Goal is to determine relationship that takes as many factors as possible into account Linear Relationship Assumptions: Sun’s gravitational effect is negligible Moon’s gravitational pull is constant •Moon’s orbital speed is constant •Moon’s distance from Earth is constant Increase in Earth’s day as a function of time currently has approximate value of 0.0016 seconds/century. Equation can be modeled linearly: D(t) = 86400 s + (0.0016 s/century)(t centuries) D(t) => length of day as function of time centuries 86400 s => Current length of day 0.0016 => Constant increase in day length s/century Solution: Simply set D(t) = 28.5 days 28.5 days = 2,462,400 s 2,462,400 s = 86400 s + (0.0016 s/century)(t centuries) Solve for t 2,462,400 s = 86400 s + (0.0016 s/century)(t centuries) 2,376,000 s = (0.0016 s/century)(t centuries) 1,485,000,000 centuries = t T = 148,500,000,000 years 148.5 billion years “To Assume Makes…” Linear Equation assumes that force of gravity is constant Untrue; gravitational force varies with distance as expressed by F = gmM/r2 Moon is drifting away from Earth by 3-4 cm each year 3-4 cm adds up over the years Must the Change in the Moon’s Gravitational Pull be Included? Yes. Current gravitational Force exerted by Moon: F = G(7.35x1022 kg)(5.97x1024 kg) / (384x106 m)2 = 1.98x1020 N Linear equation dictates that if gravitational force is constant, it will take 928.125 billion years to achieve tidal locking Check change in r for that length of time at 4 cm/year r = 384 x 106 m Δr m = .004 m/year x 148.5 x 109 years = 5.94 x 108 m r at tidal locking = r + Δr = 384 x 106 m + 5.94 x 108 m = 978 x 106 m Moon’s gravitational pull F at tidal locking: F = GmM/r2 = G(7.35x1022 kg)(5.97x1024 kg) / (978 x 106 m)2 = 3.06 x 1019 N Calculating a Complex Equation Assumptions: Sun’s gravitational effect is negligible Moon’s gravitational effect is NOT constant Change in Earth’s rotational speed (and thus increase in length of day) is dependent on Gravitational force of Moon. D(t) is related to strength of Moon’s Gravitational pull F. F is related to distance between Earth and Moon by F = GmM/r2 r is changed by 3-4 cm (.003 m to .004 m) each year r = 384 x 106 meters today Assume maximum change in Moon’s distance from Earth (.004 meters/year) r(t) meters = 384 x 106 meters + (.004 meters/year) (t years) F = GmM/r2 F = GmM/(r(t))2 F = GmM/(384 x 106 meters + (.004 meters/year) (t years)) 2 The Next Step Find equation that relates change in Earth’s rotational speed to gravitational force exerted by Moon Another Problem: The Conservation of Angular Momentum Looking at Earth-Moon as isolated system Gravity is only force acting between bodies Gravity is a conservative force Need to determine where energy from Earth’s rotation is transferred to (if Earth slows down, something else must change/speed up to compensate) A Case Study? Need to determine which factors are in play and how they effect equations and results. Factors: Moon’s drift from Earth, Moon’s rotational speed, Moon’s orbital speed. Case 1 Potential energy gained from increase in Moon’s distance from Earth accounts for all energy transferred from Earth’s rotation If Moon’s 3-4 cm drift per year (and thus increase in gravitational potential energy) accounts for change in Earth’s rotational speed, equation expressing the Moon’s change in orbit period must be used for solution Simple solution; change will be a linear equation (assuming 3-4 cm drift is constant) Case 2 Potential energy gained from increase in Moon’s distance from Earth and change in Moon’s rotational speed accounts for energy transference Same solution as Case 1, since change in rotational speed will not effect gravitational force of Moon Case 3 Potential energy gained from increase in Moon’s distance from Earth and change in Moon’s orbital speed accounts for all energy transference Moon’s orbital period will have to be expressed as a function That function will have to be based on function that expresses the change in both the orbital distance and the orbital speed of the Moon Case Argh Some variable combination of the previous three cases Painful. Things I Refuse To Look At Right Now Effect that Earth’s change in axial tilt has on the Moon’s gravitational pull on the Earth Non-circular orbits The gravitational effect of the Sun My To-Do List Determine function that relates change in Earth’s rotational speed to change in Moon’s gravitational force Determine which cases are realistic and solve for each Hope that this is actually calculable Questions/Comments/Insults? Sources: Tides S J Peale Encyclopedia of Astronomy & Astrophysics © IOP Publishing Ltd 2006 ISBN 0333750888 Physics For Scientists and Engineers 3rd e Douglas C Giancoli © Prentice Hall 2000 ISBN 0-13-021517-1 Universe 7th e Roger A Freedman William J Kaufmann III © W.H. Freeman and company 2005 ISBN 0-7167-9884-0