ELLIPTICAL ORBITS & VENUS

Transcription

1 ELLIPTICAL ORBITS & VENUS Problem set 2 due now on the front table 18 September 2018 ASTRONOMY 111 FALL TIDAL LOCKING AND MERCURY S ORBIT In celestial mechanics, tidal locking means that the heat dissipated during an orbit, by variations in the tidal forces, is minimized. Tidal forces are the gravitational force differences between one end and another of a body due to different distances away from a second body, and from one side to another of a body due to different directions toward the second body. The result is a stretch along the line between the two bodies and a compression in the perpendicular directions. We will derive formulas for tidal forces and much more on tides later this semester. 18 September 2018 ASTRONOMY 111 FALL

2 AN INTRODUCTION TO TIDAL FORCES Earth pulls all things toward its center, and pulls harder on things which are closer. To you, it seems as if you are stretched in the direction of the Earth s center and squeezed in the other direction. Figure from Thorne, Black Holes and Time Warps 18 September 2018 ASTRONOMY 111 FALL TIDAL LOCKING & THE MOON S ORBIT A body in circular orbit, with rotation period equal to revolution period, suffers no change in tidal forces as it travels its orbit. Thus, there is no heat dissipated from stretching and shrinking. The Moon s orbit about Earth has a fairly small eccentricity (0.055, so it can be considered circular to good approximation. The Moon s revolution and revolution periods are identical, so it is tidally locked. 18 September 2018 ASTRONOMY 111 FALL

3 MERCURY S ELLIPTICAL ORBIT Mercury s orbit has a much larger eccentricity ( and thus is more distinctly elliptical. Reminder of the properties of ellipses: Equation! " " + %" & " = 1 Foci at x = ±c = " & " Eccentricity + = = " & " Focal lengths (aphelion and perihelion distances, = ± = 1 ± + f 1 f 2 * * c a b 18 September 2018 ASTRONOMY 111 FALL ORBIT COMPARISON ε = 0 (Circle ε = (Mercury Drawn to scale, in units of semimajor axis length (a; coordinate axis origin on one focus. ε = (Moon ε = September 2018 ASTRONOMY 111 FALL

4 MERCURY S ORBIT Aphelion Since the distance between the Sun and Mercury varies by so much through an orbit, it is not possible to have no change in tides. Nevertheless, astronomers used to think that Mercury s rotation and revolution periods were equal (at 88 days because that was what observations seemed to indicate and because it agreed with naïve expectations based on circular orbits. a = AU f 2 = AU Sun f 1 = AU b = AU Perihelion 18 September 2018 ASTRONOMY 111 FALL MERCURY S ORBIT In the mid-1960s, though, radar astronomers at Arecibo Observatory showed that the rotation period of Mercury is actually closer to 59 days. Shortly thereafter, theoreticians realized that the minimum in tidal heating for Mercury s eccentric orbit was reached for an orbital period exactly 3/2 times the rotation period. In the mid-1970s, the fly-bys of Mariner 10 yielded the rotation period much more precisely and bore out this prediction. Mercury rotates exactly three times every two orbits. 18 September 2018 ASTRONOMY 111 FALL

5 MERCURY S ORBIT Like the Moon, the tidal forces to which it is subjected has made Mercury slightly oblong. The orbit is arranged so that the long axis always points toward the Sun at perihelion. This keeps the tidal stretching to a minimum, though the minimum is not zero. Other trivia: the perihelion of Mercury s orbit precesses slowly in a prograde direction. Long a mystery, the precise accounting for this orbital precession by warping of space by the Sun s gravity was one of the first triumphs of Einstein s general theory of relativity ( September 2018 ASTRONOMY 111 FALL MOMENT OF INERTIA AND PLANETARY DIFFERENTIATION For a given spin rate, different distribution of a given mass within the bounds of a planet would lead to different values of a planet s spin angular momentum. The relation between total angular momentum L and angular frequency ω is! = $ where = % & ' ( * The moment of inertia can be measured from the fine details of a planet s gravitational field, as we will discuss in detail later this semester. x z & ' = & sin. dm θ r φ y 18 September 2018 ASTRONOMY 111 FALL

6 MOMENTS OF INERTIA Here are some simple spherical configurations of mass all with mass M and radius R and the corresponding moment of inertia. Constant-density sphere:! = $ %& Spherical shell with constant mass per unit area:! = ' %& Density maximum at center, declining linearly to zero at r = R:! = ( $ %& = 0.267%& The more strongly concentrated the mass is at the center, the smaller the leading factor in the expression for I. 18 September 2018 ASTRONOMY 111 FALL MOMENTS OF INERTIA & PLANETARY DIFFERENTIATION The Moon has I = 0.394MR 2, close enough to! = $ %& that its density must be nearly constant. Mercury has I = 0.33MR 2, significantly smaller than the uniform-density value and probably reflective of a much larger density in the center than at the surface. An iron core, of the sort described above, would satisfy this. The moment of inertia and the average density can be used to calculate the properties of the core fairly confidently. A planet that is not concentrated at the center but tends to have a uniform density is called undifferentiated. The process of concentration of heavier minerals and metals at the center is called differentiation. 18 September 2018 ASTRONOMY 111 FALL

7 VENUS Inertial moments for spherical bodies: detecting differentiation Venus as a twin of Earth: rock composition, tectonics, volcanism, and differentiation Melting the insides of infant terrestrial planets with radioactivity Loss of water and organic molecules from hot infant planets 2012 transit of Venus (Hinode, JAXA/NASA/Lockheed Martin 18 September 2018 ASTRONOMY 111 FALL MOMENT OF INERTIA Angular momentum L and moment of inertia I of a point mass m revolving in a circle with radius r :! = $% & = % & ' $ % & = ( r ( = % & ' Units are g cm 2 = +, - Units are rad/s In general, for a point mass m with velocity v, a displacement r from the coordinate origin:. = / 1 = / 2 = $% sin 6 78 = $% & 78 r r m 78 α v v 18 September 2018 ASTRONOMY 111 FALL

8 BREAKING A SPHERE INTO MANAGEABLE BITS For non-point masses, e.g. planets, one finds the moment of inertia by decomposing the mass into chunks small enough to treat as point masses (infinitesimal elements: dm = ρ dv, and then add up (integrate their moments of inertia. z In Cartesian coordinates:!" =!$!%!& In spherical coordinates:!" =!' '!( ' sin (!, θ r r sinθ dφ dr r dθ dz dy dx = ' - sin (!'!(!, x φ y 18 September 2018 ASTRONOMY 111 FALL !, +,, ', ', *! = sin ' cos * + = sin ' sin *, = cos '!, =! , - ' = tan 12! , * = tan 12 +! + * REMINDER: THE SPHERICAL COORDINATES R, Θ, Φ 18 September 2018 ASTRONOMY 111 FALL

9 THE AST 111 CALCULUS PALETTE, PAGE 1 Fundamental theorem of calculus: if! " = $ $% & ", then (! " *" = & + & - $ $% ". = /".01 ( ". *" = 3 % =.71 ( $ $% 89% = :8 9% ( 8 9% *" = ( $ sin :" = : cos :" 1 $% ( cos :" *" = sin :+ sin :- 9 $ cos :" = : sin :" $% 1 ( sin :" *" = cos :+ cos : September 2018 ASTRONOMY 111 FALL MOMENT OF INERTIA OF AN UNDIFFERENTIATED SPHERE ABOUT AN AXIS THROUGH ITS CENTER Suppose the sphere has mass M, radius R, and uniform density ρ, by which we mean that the density does not depend on r, θ, or φ. Then we know the relation of M and ρ without doing an integral:! = $ = 3 4'( Without the sphere, each bit of mass dm follows a circle with radius * + = * sin /: 01 = * = * sin / =! 0$ =!* 2 sin / 0* 0/ 04 x φ z * + = * sin / dm θ r y 18 September 2018 ASTRONOMY 111 FALL

10 MOMENT OF INERTIA OF AN UNDIFFERENTIATED SPHERE Thus! = $! = % & ' $( z '* * + = % sin / ' 0% ' sin / $% $/ $1 % & = % sin / dm = 0 '* $1 * + sin / 2 $/ % 3 $% Three one-variable integrals we can take one at a time: '* + x θ r φ y $1 = 25 % 3 $% = September 2018 ASTRONOMY 111 FALL MOMENT OF INERTIA OF AN UNDIFFERENTIATED SPHERE The third one takes a few more steps:! " " sin ' ( ' =! 1 cos ' / sin ' ' =! sin ' ' +! cos ' / sin ' ' Substitute: 1 = cos ', 1 = sin ' ' As ' = 0 4, 1 = 1 1, so! " sin ' ( ' = cos ' 5 +! " 6 < = =24 4 >? 3 76 " 1 / 1 = " 1 ( > ( 24 4 >? 3 5 = 2 5 A>/ 5 = 3A = = September 2018 ASTRONOMY 111 FALL

11 MOMENT OF INERTIA OF A DIFFERENTIATED SPHERE For example: density decreasing linearly from a central value of ρ 0 to zero at the surface:! =! 1 & ' This time, we have to do an integral to get M and ρ 0 in terms of each other: ( =! & & * sin. (& (. (/ 0 = 1 ( =! 1 *2 (/ 1 2 sin. (. 1 ' 8 3 & * 1 & ' (& M =! '8 = '8!! = 30 7' 8 18 September 2018 ASTRONOMY 111 FALL MOMENT OF INERTIA OF A DIFFERENTIATED SPHERE! = $ % & '( &* * + = $ sin / & 0 1 $ 3 $& sin / '$ '/ '4 &* = 0 = '4 3 : * 5 3: 6 + sin / 5 '/ $ 6 1 $ 3 '$ 3 : = The integrals over θ and φ are the same as before (4/3 and 2π and will be as long as the density does not depend on an angle. = 4 15?3& = 0.267?3 & 18 September 2018 ASTRONOMY 111 FALL

12 VENUS VITAL STATISTICS Mass x10 27 g (0.815M C. Hamilton (Galileo/JPL/NASA Equatorial radius x10 8 cm (0.949R Average density g cm -3 Moment of inertia 0.33MR 2 Albedo 0.67 Orbital semimajor axis x10 13 cm ( AU Orbital eccentricity Sidereal revolution period Sidereal rotation period Length of day Magnetic field days days days 18 September 2018 ASTRONOMY 111 FALL VENUS ATMOSPHERE S VITAL STATISTICS Surface pressure: 92 earth atmospheres Average temperature near surface: 737 K (464 C Compare to T = 327 K expected for its orbit Diurnal (day-night temperature range: ~0 Surface wind speeds: m/s Atmospheric composition (near surface, by volume: 96.5% CO 2, 3.5% N 2, 0.015% SO 2, 0.007% Ar, 0.002% H 2 O Ultraviolet image of Venus cloud tops by the Akatsuki (JAXA 18 September 2018 ASTRONOMY 111 FALL

13 PHASES OF VENUS At most, Venus gets 47 from the Sun as noted by the ancients and displays phases, as first noted by Galileo (1610. These facts directly show that Venus orbits the Sun, not the Earth. Photo by Wah! 18 September 2018 ASTRONOMY 111 FALL VISITS TO VENUS Venus was the first planet to receive an Earthly spacecraft on its surface: the Soviet Venera 7, in 1970, lasted 23 minutes and made the first in situ report of temperature and pressure (748 K, 90 atmospheres. Including missions with other final destinations, there have been 27 successful visits between Mariner 2 (1962 and the current JAXA Akatsuki (2010-present, highlighted by many Soviet Venera landers. Also a large number of failures, mainly in the Soviet Sputnik, Zond, and Cosmos series. Venera 9 landing site: the first image retrieved from the surface of a planet besides the Earth-Moon system (Ted Stryk 18 September 2018 ASTRONOMY 111 FALL

14 FAILED VISITS TO VENUS Failures are dominated by early Soviet losses at launch or en route. The success rate of landers is good, considering the high degree of difficulty descending through the dense atmosphere and operating in extreme heat. Agency Successful Venus missions Unsuccessful Venus missions NASA (USA 10 1 USSR ESA (EU 1 0 JAXA (Japan 1 0 Agency Successes in retrieving data from the surface Failures in retrieving data from the surface Venera and Vega (USSR 10 4 NASA (USA September 2018 ASTRONOMY 111 FALL THE COMPOSITION OF VENUS SURFACE Five of the Venera landers made composition measurements on soil and/or rock. Venera 8 reported a composition similar to granitic (felsic rocks; all the others obtained a mafic composition. The best measurements come from Venera 13 and 14, which landed in the highlands (13 and lowlands (14 of Beta Regio and drilled into solid rock. Both measured compositions similar to alkaline basalts, similar to those retrieved from mid-ocean ridges on Earth and distinctively different from Lunar mare basalts. X-ray spectra of Venereal rocks (solid curves by Venerai13 (top and 14 (right compared to ocean-floor basalts (points (Surkov et al September 2018 ASTRONOMY 111 FALL

15 9/18/18 THE STRUCTURE OF VENUS SURFACE Most mapping of Venus has been made by (cloud-penetrating radar maps, such as those by the NASA Magellan and ESA Venus Express orbiters. Venus Express also includes high resolution infrared cameras capable of thermal imaging and shows the warmest (coolest parts of the surface lie at the lowest (highest elevations. VIRTIS/Venus Express (ESA 18 September 2018 ASTRONOMY 111 FALL ASTRONOMY 111 FALL VENUS SURFACE STRUCTURE The radar maps reveal many of the features of plate tectonics and volcanism that we know on Earth: Many deep, asymmetrical troughs that resemble subduction zones, and long ridges that resemble mid-ocean ridges Some large shield volcanos and unique lava vents such as coronae and pancake domes Lava flows are seen to stretch many hundreds of km from such volcanoes. Hardly any impact craters Magellan/JPL/NASA 18 September

16 VENUS SURFACE STRUCTURE Most of this activity is extinct, though. The tectonic-like features do not link up to make a global plate boundary system as on Earth. Still has active volcanoes, but not very many. Overall, there has not been much tectonic activity over the past few hundred million years, although there was some long ago. Magellan/JPL/NASA 18 September 2018 ASTRONOMY 111 FALL VENUS INTERNAL STRUCTURE Like Mercury and Earth, Venus is differentiated: It has a large bulk (average density (5.243 g cm -3 but its surface is covered in mafic silicate rocks (3.2 g cm -3 It has a rather small moment of inertia (I = 0.33MR 2, too small to represent uniform density (I = 0.40MR 2. As one might suspect a priori, Venus internal structure appears similar to Earth s, which we think includes a liquid iron-nickel core, plastic mantle, and crust. Diagram by UCAR (U. Michigan 18 September 2018 ASTRONOMY 111 FALL

17 WHY NEW PLANETS ARE MOLTEN AND THUS BECOME QUICKLY DIFFERENTIATED Brand-new terrestrial planets are usually very hot, hot enough for most or all of the volume to be molten. If the interior is molten, buoyancy rules: dense metals and minerals can sink freely to the center, and lighter ones can float to the surface. There are two reasons why the interiors are so hot. Accretion. The gravitational potential energy of a planet s ingredients decreases from zero to very large negative values in the act of forming, and since energy is conserved, this leads to very large positive energy in the form of heat. Radioactivity. Just as in nuclear reactors, it converts nuclear electrostatic potential energy to kinetic energy of the products, which heat the medium. And planetary ingredients were a lot more radioactive then than now. 18 September 2018 ASTRONOMY 111 FALL WHY NEW PLANETS ARE MOLTEN We will not be discussing accretion, heat transport, or the temperatures of planetary interiors until next month. But we can get an appreciation for the temperatures using what we now know about the surface temperatures and radioactivity: Radioactivity occurs throughout the interior, and its heat input can be calculated from just the mass if the composition is typical of the materials out of which the planets were built:! rad = &Λ rad ( The heating rate per gram, Λ rad, decreases with time since there is no means to replenish decayed radionuclides with fresh material from the interstellar medium. In turn, this heat input can just be added to that due to sunlight to work out the temperature of the surface. 18 September 2018 ASTRONOMY 111 FALL

18 WHY NEW PLANETS ARE MOLTEN Radioactive heating rates for primitive (carbonaceous chondrite meteoritic material, currently and at the birth of the Solar System. Radionuclide Daughter(s Half-life [Myr] Heating today [10-8 erg/s/g] 40 K 40 Ar, 40 Ca Th 208 Pb U 207 Pb U 206 Pb Al 26 Mg Cl 36 S, 36 Ar Heating 4567 Myr ago [10-8 erg/s/g] 60 Fe 60 Ni Λ rad(t for primitive meteorites [erg/s/g] 5.52x x September 2018 ASTRONOMY 111 FALL RADIOACTIVE HEATING Consider the surface temperature T s for a spherical planet (mass M, radius R of solar-system composition a distance r from a star with luminosity L.! " sunlight and radioactive heating =! 456"789 +! ;<= =! >59 blackbody radiation B 4DE F DGF + HΛ ;<= J = 4DG F KL M N L M = BGF + 4HΛ OPQ J E F 16DKE F G F T/N The Sun s luminosity was larger by a factor of about 2.5 back then, but Λ rad was larger by a factor of almost one million. 18 September 2018 ASTRONOMY 111 FALL

19 RADIOACTIVE HEATING Results, assuming that each body lacks an atmosphere and formed in a time short compared to the half-life of 60 Fe. Compare to the melting points of planetary ingredients: K for silicate rocks, 1800 K for iron. And, though we have not learned how to calculate interior temperatures yet, most of us will realize that each body is hotter inside, if heated from within. Venus was almost certainly molten all the way through when new, and probably Mercury as well. Planet T s now [K] T s 4567 Myr ago [K] Venus Mercury Moon September 2018 ASTRONOMY 111 FALL LOTS OF VOLATILE MOLECULES FROM NEWBORN PLANETS Venus original ingredients were no doubt similar to the small primitive Solar-system bodies we find as meteorites and were rich in organic molecules and ices. These molecules contain nearly all the carbon, nitrogen, and hydrogen in meteorites. A little oxygen, too. When Venus was molten, these molecules were all evaporated into a temporary atmosphere. Ultraviolet sunlight quickly breaks such molecules up into atoms, which are all light enough to escape the gravity of the planet. Thus, Venus, like every other terrestrial planet, is very poor in carbon, nitrogen, and hydrogen, compared to other Solar System bodies. 18 September 2018 ASTRONOMY 111 FALL

20 LOSS OF VOLATILE MOLECULES The carbon, nitrogen, and hydrogen currently possessed by the terrestrial planets were added by captured asteroids and comets after their surfaces cooled. This includes the hydrogen in water: 90% from asteroids, 10% from comets. Element Earth Sun Comets Hydrogen Carbon < Nitrogen Oxygen Sodium Magnesium Aluminum Silicon Phosphorus 0.03 Sulfur Potassium 0.02 Calcium Iron Nickel September 2018 ASTRONOMY 111 FALL