Lecture 19: The Moon & Mercury The Moon & Mercury The Moon and Mercury are similar in some ways They both have: Heavily cratered Dark colored surfaces No atmosphere No water They also have some interesting differences: The Moon is much smaller than Mercury Mercury has a much larger core They have different formation histories The Moon & Mercury 1
The Moon The Moon is the only natural satellite of the Earth It has no atmosphere, no water, and no sound. The Moon is completely desolate With no atmosphere to moderate the surface temperature, the daily variation is extreme: T noon 400 K (above boiling point of water) T night 100 K (below freezing point of water) The surface of the Moon has been altered by thousands of meteorite and asteroid impacts The crater evidence has been preserved, since there is no wind or water erosion Motion of the Moon The Moon revolves around the Earth with a period of approximately one month It keeps the same face pointed towards Earth at all times The Moon s orbit is not in the same plane as Earth s orbit around the Sun (it is tilted by 5.2 o ) Therefore, eclipses are infrequent Earth Sun Moon Orbit of Moon The Moon The lunar orbit is not perfectly circular the eccentricity is e 0.055 The average Earth-Moon distance is 384,000 km (237,000 miles) The perigee distance is 363,000 km (closest to Earth) The apogee distance is 405,000 km (farthest from Earth) 2
The Moon The sidereal period of the lunar orbit is 27.3 Earth days This is one sidereal month, measured relative to the stars The synodic period of the Moon s orbit is 29.5 Earth days 1 synodic month later Distant Star Sun Moon 1 sidereal month later Earth Orbit of Moon starting point The Moon Hence 29.5 Earth days is one synodic month, measured relative to the Sun The synodic period is the time between full s, or new s, etc. 1 synodic month later Distant Star Sun Moon 1 sidereal month later Earth Orbit of Moon starting point 3
Phases of the Moon Eclipses Can occur only if the alignment is precise (Earth, Moon, and Sun line up exactly) The Moon s orbit is tilted by 5.2 o relative to the ecliptic Therefore, the Moon, Earth, and Sun must lie along the Line of Nodes, which is the intersection of the two orbital planes Eclipses can be Lunar (Moon becomes darker) or Solar (Sun becomes darker) Line of Nodes Solar Eclipse Sun 4
Eclipses Eclipses 5
Tidal Locking The Moon always keeps the same face pointed towards the Earth The spin period relative to the stars is exactly equal to the orbital period, 27.3 Earth days Since these periods are equal, we say that the Moon is tidally locked, in a synchronous orbit around the Earth Tidal Force Tidal Force The Moon is in a synchronous orbit, but the Earth is not. Why is this? Because: The tidal force of the Moon on the Earth is 2GM F earth M D earth 3 earth The tidal force of the Earth on the Moon is 2GMearthMR F 3 D The tidal force exerted on the Moon by the Earth is therefore 20 times larger than that exerted on the Earth by the Moon R 6
Tidal Locking The tidal force exerted on the Moon by the Earth is 20 times larger than that exerted on the Earth by the Moon The Moon is 80 times less massive than the Earth and therefore the effect of the tidal force is much stronger The tidal force of the Earth causes a tidal bulge in the crust of the Moon Tidal Heating In the distant past, the Moon did spin relative to the Earth, and that caused the tidal bulge to ripple around the Moon, like the tides on Earth The motion of the tidal bulge on the Moon caused distortion of the rocks, leading to frictional heating of the lunar crust Tidal Heating The heating caused the gradual dissipation of the lunar spin. The kinetic energy of the Moon s spin has been used to heat the lunar crust The same effect is gradually slowing down Earth s spin Conservation of angular momentum is causing the Moon to move away from the Earth 1-2 cm per year 7
Tidal Locking Eventually, the Moon with have an average orbital distance of about 550,000 km (compared with the current average distance of 384,000 km) When this happens, the Earth will be tidally locked to the Moon, and then the Earth will therefore always show the same face towards the Moon This will take about 10 12 years to happen The Moon The near side of the Moon includes heavily cratered highlands and less cratered lowlands, or maria The maria (seas) are dark colored, flat regions created by ancient lava flows The highlands (terrae) are bright areas elevated several kilometers above the maria highlands maria The Moon Since the Moon is in synchronous orbit around the Earth, the far side of the Moon was completely unknown until it was first explored by spacecraft in the 1960 s 8
The Moon The far side of the Moon is more heavily cratered, and contains a smaller amount of dark maria This suggests that the crust is thicker on the far side, and the molten material forming the maria was pulled towards the Earth The tidal force of the Earth may have caused this while the Moon was much more molten than it is now The Moon Geologically and biologically, the Moon is dead: No life No volcanoes No internal geological activity However, there are quakes that occur when meteorites impact on the surface Moonquakes have been detected by seismometers left on the Moon by the Apollo astronauts These seismometers sent back radio signals to Earth until 1977 They helped us study the Moon s interior structure Mercury Mercury s orbit is elliptical, with eccentricity e 0.2056 The average Sun-Mercury distance is 0.39 AU (58,766,000 km) The perihelion distance is 0.31 AU (closest to Sun) The aphelion distance is 0.47 AU (farthest from Sun) Mercury always appears close to the Sun in the sky 9
Mercury The angle between Mercury and the Sun (as viewed from Earth) is called the elongation Based on the size and shape of Mercury s orbit, we find that Elongation 18 o (perihelion) Elongation 28 o (aphelion) Hence Mercury never appears very far from the Sun as viewed from Earth Mercury & the Sun Mercury Mercury can be viewed with the naked eye only when the Sun s light is blocked Just before dawn Just after sunset During total solar eclipse Sunrise: 10
Mercury Why is Mercury visible for no more than two hours? The Earth rotates 360 o in 24 hours, or o o 360 15 24 hr hr Since the maximum elongation of Mercury is 28 o, the maximum time for the Sun to rise after Mercury is o 28 t 2 hours o 15 / hr Mercury Mercury shines by reflected sunlight, just like the Moon With the naked eye, only the first or third quarter phases are visible Using large telescopes, Mercury can be viewed even during the daytime by filtering out the sunlight Mercury Messenger website Phases of Mercury 11
Moon Mercury The albedo of an object is the fraction of the incident light that is reflected Albedo 0.1 for Mercury Albedo 0.1 for Moon Albedo 0.4 for Earth Albedo 0.7 for Venus Even though Mercury s albedo is so low, it appears very bright because it is so close to the Sun Mercury s Orbit The semi-major axis of Mercury s orbit is a 0.39 AU Mercury Sun Kepler s third law relates the semi-major axis to the orbital period a 12
Mercury s Orbit Kepler s third law relates the semi-major axis a to the orbital period P 2 P years Solving for the period P yields 3 a AU P a years AU 3/ 2 Since a 0.39 AU for Mercury, we obtain P 0.244 Earth years or P 88 Earth days Mercury s Orbit Since Mercury is closer to the Sun than the Earth is, it s apparent motion can be retrograde Transits occur when Mercury passes in front of the solar disk as viewed from Earth This happens about once every 10 years Mercury (1 hour increments) Bulk Properties of the Moon It is interesting to compare the bulk properties of the Moon and Mercury We have for the radius and mass of the Moon R 1,700 km 0.25 R earth M 7.4 x 10 25 g 0.0125 M earth The volume is therefore given by V Hence V 2 x 10 25 cm 3 4 π 3 3 R 13
The average density of the Moon is therefore ρ Bulk Properties of the Moon M V 25 7.4 10 g 25 3 2 10 cm We obtain ρ 3.5 g cm 3 This is similar to the density of the surface rocks in the Earth s crust The Moon does not contain much iron, or a large, dense core Gravity on the Moon We can compute the surface acceleration for the Moon using Newton s laws: - GMm F ma 2 R Solving for the surface acceleration A yields - GM A 2 R Gravity on the Moon Using the lunar values for the mass and radius, we obtain for the surface acceleration A - GM R 2 1.7 ms 2 For the Earth we the surface acceleration obtained is A earth - GM R earth 2 earth 9.8ms 2 14
Gravity on the Moon The ratio of the lunar and Earth accelerations is A A 1 6 Hence objects on the Moon weigh about 1/6 of their weight on Earth Example: jump by 180 lb astronaut, who weighs 30 lbs on the Moon earth 15
Bulk Properties of Mercury We have for the radius and mass of Mercury R mercury 2,439 km 0.38 R earth M mercury 3.3 x 10 26 g 0.05 M earth The volume is therefore given by V 4 π 3 3 mercury R mercury Hence V mercury 6.1 x 10 25 cm 3 The average density of Mercury is therefore ρ Bulk Properties of Mercury mercury M V mercury mercury 26 3.3 10 g 25 6.1 10 cm 3 We obtain ρ mercury 5.4 g cm 3 This is similar to the average density of the Earth Hence Mercury probably contains a large, dense core Comparison with the Earth The average density of the Earth is equal to ρ Θ 6 g cm 3 The density of water is ρ water 1 g cm 3 The density of rock is ρ rock 2 4 g cm 3 16
Gravity on Mercury Using Mercury s values for the mass and radius, we obtain for the surface acceleration A merc - GM R merc 2 merc For the Earth we have found that The ratio of the Mercury and Earth surface accelerations is therefore A A A merc earth 3.7 ms earth 9.8ms 0.38 2 2 Hence objects on Mercury weigh about 1/3 of their weight on Earth 17
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