The Lunar Orbit Throughout Time and Space Adrienne Dove Stuart Robbins Colin Wallace September 2005 ASTR 5835 Planetary Seminar 1 What We See Today 2 To First Order The Moon Orbits the Earth Actually the Moon orbits the Sun and is perturbed by the Earth This orbit is elliptical but the shape is changing with a varying eccentricity recessing nodes and precessing apsides The orbit is also inclined 5 with respect to the ecliptic 3 The Orbit of the Moon and the Wobble of Earth The center of mass of the Earth Moon system is given by M R1 M moon R2 R1 M moon M R2 moon R R1 M M M moon R1 R M M moon This lies 1800 km below Earth s surface thus the Earth wobbles 4 Longitudinal Librations in Action While the Moon s rotation rate may be constant the fact that its orbit is slightly eccentric means that its orbital velocity changes This causes longitudinal librations 5 The Moon s Orbit is Inclined 5 to the Ecliptic This results in latitudinal librations For completeness there are also diurnal librations 6 APOD http antwrp gsfc nasa gov apod ap051113 html One Lunation 7 Lunar Nodes The two points where the lunar orbit intersects the ecliptic are called nodes The location of a node determines whether or not an eclipse will occur The Moon s orbit twists over time and the position of the nodes changes The nodes recess over a period of 18 6 years 8 Moon s Orbital Path The Moon s orbit with respect to the Sun looks like this The Moon s orbit with respect to the Sun does NOT look like this Conclusion The Moon s orbit with respect to the Sun is convex But why 9 Moon s Orbital Path The Moon s orbit with respect to the Sun looks like this The Moon s orbit with respect to the Sun does NOT look like this Conclusion The Moon s orbit with respect to the Sun is convex But why The gravitational force of the Sun on the Moon is more than twice the gravitational force of the Earth on the Moon 2 GM M moon R FSM 2 21 2 FEM GM M moon Rmoon 10 Moon s Orbital Path So the net force is directed toward the Sun Force is proportional to acceleration via Newton s s e c o n d l a w The acceleration is the second derivative of position and concavity depends on the second derivative Ergo the Moon s orbit is convex with respect to the Sun to an observer in space the Moon must appear as a normal planet traveling in an elliptical orbit with the Sun in one of the foci Moore 2001 The moon is really orbiting the sun in a manner that is significantly perturbed by the gravitational field of the earth Hodges 2002 11 Apsidal Precession The speeding up and slowing down of the Moon s circumsolar motion relative to Earth causes the line of apsides to precess with a period of 8 85 years Evection Variation Century Max Perigee Max Apogee 20th 221 451 miles 252 731 miles 21st 221 535 miles 252 728 miles 12 How the Moon s Orbit Evolved Through Time 13 Evidence for a Changing Orbit Present Day The Experiment Apollo Lunar Laser Ranging LLR Observatories throughout the world have used these for 30 years The Results Lunar recession rate is presently 3 82 0 07 cm per year Running the results backwards the moon is impossibly young with its orbit only 1 5 Gyrs old 8 13 2 3 84 10 m 2 a13 2 2 t 0 1 5 Gyr 45 13 2 1 13 f 13 6 29 10 m yr 14 Evidence for a Changing Orbit Historical Fossil Evidence for Tidal Period Sedimentary Evidence for Tidal Period Models Based Upon Ocean Resonances Figure 1 Bills Ray 1999 15 What Governs a Changing Orbit The Picture 16 What Governs a Changing Orbit The Equation k M da 1 2 11 2 11 2 5 moon f a 3 R G M moon M a dt Q M This shows how to calculate the change in the moon s semi major axis over time To get the age of the lunar orbit value from before a constant f was assumed But this is not the case 17 What Governs a Changing Orbit The Equation k t M 11 2 da 1 2 11 2 5 moon R G M f a 3 a moon M dt Q t M This shows how to calculate the change in the moon s semi major axis over time To get the age of the lunar orbit value from before a constant f was assumed But this is not the case because the Earth ocean tidal Love number and the dissipation quality factor change though time 18 What Governs a Changing Orbit Graph Dating shows the Moon is over 4 Gyrs old k Q must have changed through time We are in an anomalously high k Q ratio today meaning that the tidal bulge on Earth is larger forcing a greater recession rate k Q can change based upon where land masses are Figure 2 Bills Ray 1999 19 Various Tidal Models Simplify Assume that orbits are simple Darwin Tides Fourier expand the tide potential and introduce friction through numerous and unconstrained phase lags MacDonald Tides Model distortion of the body as a second harmonic distortion adds friction as a delay mechanism for a constant arbitrary phase lag doesn t make sense for an eccentric orbit Mignard s Darwin Tides Simple analytic form with a second harmonic distortion but delayed relative to the tide potential by a constant time lag This is the same as Darwin s if the phase factors are proportional to frequency 20 What That Means for Where the Moon Was Maybe 21 Where Was the Moon Assume it formed from a ring of material orbiting the Earth after the Giant Impact So you would initially expect it to be orbiting in the equatorial plane of the Earth http th nao ac jp kokubo moon kit movie html 22 Calculating a Previous Orbit Calculation of the History of the lunar orbit is fraught with difficulties Wisdom AJ 2006 Difficulties Rate of dissipation Inclination Eccentricity 23 Initial Conditions Remember current values a 3 84402 108 da dt 3 82 0 07 cm yr Records indicate that 620 Mya A day was 21 9 0 4 hrs There were 13 1 0 1 synodic months per year There were 400 7 solar days per year a a0 0 965 0 005 The average recession rate was 2 17 0 31 cm yr Records indicate that 2 45 Gya There were 14 5 0 5 synodic months per year a a0 0 906 0 029 The average recession rate was 1 24 0 71 cm yr Taken together this indicates that the moon wasn t close to Earth 1 5 Gya 24 Inclination Orbital Plane is inclined 5 to the ecliptic But this doesn t match what we d expect if it had formed …