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NEAR of the 21 Lunar Landings, 19—All of the U.S Copyrights Prof Marko Popovic 2021 NEAR Of the 21 lunar landings, 19—all of the U.S. and Russian landings—occurred between 1966 and 1976. Then humanity took a 37-year break from landing on the moon before China achieved its first lunar touchdown in 2013. Take a look at the first 21 successful lunar landings on this interactive map https://www.smithsonianmag.com/science-nature/interactive-map-shows-all-21-successful-moon-landings-180972687/ Moon 1 The near side of the Moon with the major maria (singular mare, vocalized mar-ray) and lunar craters identified. Maria means "seas" in Latin. The maria are basaltic lava plains: i.e., the frozen seas of lava from lava flows. The maria cover ∼ 16% (30%) of the lunar surface (near side). Light areas are Lunar Highlands exhibiting more impact craters than Maria. The far side is pocked by ancient craters, mountains and rugged terrain, largely devoid of the smooth maria we see on the near side. The Lunar Reconnaissance Orbiter Moon 2 is a NASA robotic spacecraft currently orbiting the Moon in an eccentric polar mapping orbit. LRO data is essential for planning NASA's future human and robotic missions to the Moon. Launch date: June 18, 2009 Orbital period: 2 hours Orbit height: 31 mi Speed on orbit: 0.9942 miles/s Cost: 504 million USD (2009) The Moon is covered with a gently rolling layer of powdery soil with scattered rocks called the regolith; it is made from debris blasted out of the Lunar craters by the meteor impacts that created them. Lunar soil is composed of grains 1 cm in diameter or less. Lunar dust are particles less than 10-50 μm in diameter. The dust is electrically charged and sticks to any surface with which it comes in contact. Density of lunar regolith is about 1.5 g/cm3. Dirt becomes very dense beneath the top layer of regolith. The major processes involved in the formation of lunar soil are: Comminution: mechanical breaking of rocks and minerals into smaller particles by meteorite and micrometeorite impacts; Agglutination: welding of mineral and rock fragments together by micrometeorite-impact-produced glass; Solar wind sputtering and cosmic ray spallation caused by impacts of ions and high energy particles. These processes constitute space weathering. Lunar rocks are igneous ("fire-formed rocks"); rich in calcium (Ca), Aluminum (Al), and Titanium (Ti), poor in light elements like hydrogen (H), and with high abundance of elements like Silicon (Si) and Oxygen (O). The Maria are mostly composed of dark basalts. The Highlands rocks are largely Anorthosite. Why do we always observe the same side of the moon? Imagine Earth is fixed in space and satellite is orbiting in circular orbit… 푀 푀 3 2 푅1 < 퐺 2 푅1휔 < 퐺 2 휔 푅1 푚1 푅 < 푅 1 0 3 푀 2 푀 푅0 = 퐺 2 푅0휔 = 퐺 2 휔 푅 푅0 0 푚0 푀 푅3 > 퐺 푀 2 휔2 푅2 > 푅0 푀 2 푚2 푅2휔 > 퐺 2 푅2 Satellite consists of main body with mass 푚0 and two much smaller, rigidly connected bodies with masses 푚1 and 푚2. All three bodies have the same orbital angular speed 휔 about the Earth, mass 푀. 2 푀푚0 While the main body satisfies the stable orbital equation 푚0푅0휔 = 퐺 2 the other two do not. 푅0 푀푚 After applying 2nd Newton’s law 푚푎 − 푚휔2푅 = −퐺 one obtains 푎 푟푎푑푎푙 푅2 푟푎푑푎푙 푎푟푎푑푎푙 1 푅1 < 푅0 푅0 푀 푅2 > 푅0 푎푟푎푑푎푙 2 So far we have ignored the rigid connection between three masses. And hence, quite erroneously, it appears here that separation between masses would increase. (NEXT SLIDE) 퐹Ԧ1 푅1 < 푅0 푎1 푅0 퐹Ԧ2 푀 푎2 푅2 > 푅0 The resulting accelerations 푎1 and 푎2 of small masses 푚1 and 푚2 are perpendicular to the line connecting all three masses. The component of inertial forces parallel to that line is cancelled by attractive forces pushing masses 푚1 and 푚2 toward mass 푚0 and hence keeping distances between masses constant. Eventually the 3-body Moon reaches… Stable configuration 푀 Moon’s rotation and orbital periods are tidally locked with each other . The far side of the Moon was not seen until 1959 when it was photographed by the Soviet spacecraft Luna 3. Now you know 푀 But, we still need to explain the terminology “tidal locking”… Gravitation induced acceleration of water droplet on the surface of the Earth 푚0 퐺 2 푅0 푀 퐺 2 푚0 푅퐸 푚0 퐺 2 퐺 2 푅0 + 푅퐸 푅0 − 푅퐸 푚0 푀 푀 푀 퐺 2 퐺 2 푅퐸 푅퐸 푚0 퐺 2 푅0 Gravitation induced acceleration of water droplet on the surface of the Earth 푚 Earth is actually also orbiting about the Earth-Moon center of mass (!!!) 퐺 0 2 풎ퟎ푴 ퟐ 풎ퟎ 푅0 푮 ퟐ = 푴흎 푹ퟎ 푹 푴 + 풎ퟎ 푀 ퟎ 퐺 2 푚0 푅퐸 푚0 퐺 2 퐺 2 푅0 + 푅퐸 푅0 − 푅퐸 푚0 푀 푀 푀 퐺 2 퐺 2 Please be careful here…Earth is also orbiting due to Moon’s pull. 푅퐸 푅 퐸 풎ퟎ Hence one needs to subtract 푮 ퟐ which accounts for Earth’s acceleration… 푹ퟎ 푚0 퐺 2 푅0 Gravitation induced acceleration of water droplet on the surface of the Earth 푚0 퐺 2 푅0 푀 퐺 2 푚0 푅퐸 푚0 퐺 2 퐺 2 푅0 + 푅퐸 푅0 − 푅퐸 푚0 푀 푀 푀 퐺 퐺 푅2 2 풎ퟎ 퐸 푅퐸 Lets subtract 푮 ퟐ to accounts for Earth’s acceleration… 푹ퟎ 1 1 푅퐸 2푅0 − 푅퐸 2푅퐸 푚 퐺푚0 − = 퐺푚0 ≅ 퐺푚0 0 푅 − 푅 2 푅2 푅 − 푅 2푅2 푅3 퐺 2 0 퐸 0 0 퐸 0 0 푅0 1 1 푅퐸 2푅0 + 푅퐸 2푅퐸 퐺푚0 2 − 2 = −퐺푚0 2 2 ≅ −퐺푚0 3 푅0 + 푅퐸 푅0 푅0 + 푅퐸 푅0 푅0 Gravitation induced acceleration of water droplet on the surface of the Earth and corrected by the orbiting motion of the Earth Yes, once again, Earth is orbiting about the Earth-Moon center of mass 푀 2푅퐸 퐺 퐺푚 2 2푅 0 푅3 푅퐸 퐸 0 퐺푚0 3 푅0 푚0 푀 푀 푀 퐺 2 퐺 2 푅퐸 푅퐸 1 1 푅퐸 2푅0 − 푅퐸 2푅퐸 퐺푚0 2 − 2 = 퐺푚0 2 2 ≅ 퐺푚0 3 푅0 − 푅퐸 푅0 푅0 − 푅퐸 푅0 푅0 1 1 푅퐸 2푅0 + 푅퐸 2푅퐸 퐺푚0 2 − 2 = −퐺푚0 2 2 ≅ −퐺푚0 3 푅0 + 푅퐸 푅0 푅0 + 푅퐸 푅0 푅0 Gravitation induced acceleration of water droplet on the surface of the Earth and corrected by the orbiting motion of the Earth Τ푀표표푛 푡푑푎푙 3 푀 2푅퐸 푀 푅0 81 3 푀 퐺 2൘퐺푚0 3 = ≈ 60 ≅ 8,748,000 2푅퐸 퐺 푅퐸 푅0 2푚0 푅퐸 2 퐺푚 2 2푅 0 푅3 푅퐸 퐸 0 퐺푚0 3 푅0 푚0 푀 푀 푀 퐺 2 퐺 2 푅퐸 푅퐸 These two bulges explain why in one day there are two high tides and two low tides, as the Earth's surface rotates through each of the bulges once a day. It is surprising that this tiny moon acceleration 푀표표푛 푡푑푎푙 = Τ8,748,000 is able to create such a huge effect… For example, in the Bay of Fundy, between New Brunswick and Nova Scotia, high tides are often 12 m above low tides (!) We should have expected this result; stable configuration is ‘elongated’ along axis that connects two gravitating bodies. Tidal locking configuration Stable configuration Is there any similar effect caused by the Sun’s gravity? 3 푀 2푅퐸 푀 푅0 81 3 Recall Τ푀표표푛 푡푑푎푙 = 퐺 2ൗ퐺푚0 3 = ≈ 60 ≅ 8,748,000 푅퐸 푅0 2푚0 푅퐸 2 In the case of Sun 3 푀 2푅퐸 푀 푅푆 1 3 Τ푆푢푛 푡푑푎푙 = 퐺 2൘퐺푀푆 3 = ≈ 23,455 ≅ 19,375,000 푅퐸 푅푆 2푀푆 푅퐸 2 × 333,000 or 푆푢푛 푡푑푎푙Τ푀표표푛 푡푑푎푙 ≈ 0.45 Hence, yes, Sun’s gravity also causes tides… Moon and Sun combined But that is not all… There is an average 40 minutes delay between 푀 the Moon’s transit and the following high tide. (!) 퐺 2 푅퐸 2푅퐸 표 퐺푚0 3 10 푅0 푚0 푀 푀 푀 2푅퐸 퐺 퐺 퐺푚0 2 2 푅3 푅퐸 푅퐸 0 If this is the northern hemisphere Moon is orbiting counterclockwise. Imagine that this is the northern hemisphere Bulge is pushed “forward” due to the Earth’s -> Earth is rotating counterclockwise. rapid rotation and viscous forces. Tidal Braking This is tidal braking that slows down Earth’s rotational angular momentum while it 푀 increases Moon’s orbital angular momentum. 퐺 2 Sum of Earth’s rotational angular momentum and 푅퐸 2푅퐸 표 Moon’s orbital angular momentum is constant. 퐺푚0 3 10 푅0 푚0 푀 푀 푀 2푅퐸 퐺 퐺 퐺푚0 2 2 푅3 Moon is orbiting counterclockwise. 푅퐸 푅퐸 0 Increased Moon’s orbital angular momentum means that the Moon’s distance must increase too. Earth is rotating counterclockwise. Moon is receding from the Earth at a rate of ퟒ 풄풎/풚풆풂풓 And the duration of Earth’s day increases by 0.0016푠 every century. 휃 (Please note >0 outward and <0 inward) G푀 푚Τ2 푚 푚 − = − 휔2 푅 + 0.5퐻 cos 휃 + 푎 To understand this, we will 푅 + 0.5퐻 cos 휃 2 2 2 푟푎푑푎푙 chop the astronaut into two Τ G푀 푚Τ2 halves. G푀 푚 2 퐺푀푚/2 푚 − 2 = − 3 푅 + 0.5퐻 cos 휃 + 푎푟푎푑푎푙 푅 + 0.5퐻 cos 휃 2 푅 + 0.5퐻 cos 휃 푅 2 2 푅 푚Τ2 퐺푀푚/2 1 퐻 cos 휃 푚 퐺푀푚 − − 1 + = 푎푟푎푑푎푙 H 푚휔2푅 = 푅2 퐻 cos 휃 2 2푅 2 2 1 + 푚Τ2 푅 2푅 G푀 푚Τ2 푅 − 0.5퐻 cos 휃 2 퐺푀푚/2 4퐻 cos 휃 푚 2퐺푀퐻 cos 휃 푚 − − = = 푎 푅2 2푅 2 푅3 2 푟푎푑푎푙 푚 2퐺푀퐻 cos 휃 2 푚 2퐺푀퐻 cos 휃 sin 휃 → Stretching inertial force & Rotating inertial force 2 푅3 2 푅3 휃 (Please note >0 outward and <0 inward) G푀 푚Τ2 푚 푚 − = − 휔2 푅 − 0.5퐻 cos 휃 + 푎 To understand this, we will 푅 − 0.5퐻 cos 휃 2 2 2 푟푎푑푎푙 chop the astronaut into two Τ G푀 푚Τ2 halves.
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