2. Orbital dynamics and tides


 August Brown
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1 2. Obital dynamics and tides 2.1 The twobody poblem This efes to the mutual gavitational inteaction of two bodies. An exact mathematical solution is possible and staightfowad. In the case that one body is much moe massive than the othe (e.g., the Sun compaed to the planets) and the objects ae gavitationally bound to each othe, then the less massive body obits the moe massive one in an elliptical obit as descibed by Keple s laws. If the objects ae not gavitationally bound, i.e., the kinetic enegy is equal o lage than the gavitational potential enegy, then the tajectoy is a paabola o hypebola instead of an ellipse. In the geneal case of compaable masses, Newton s laws can be used to educe the poblem to an equivalent Kepleian situation Review of Keple s laws Keple s laws wee intoduced in the Geophysics couse, but it is woth biefly eviewing them hee (see the book by Lowie fo moe details). They descibe the obital motion of an object (e.g., planet) aound a much moe massive object (e.g., a sta). They can also be applied to moons obiting planets, etc. 1. Each planet moves in an elliptical obit about the Sun, with the Sun at one focus of the ellipse. An ellipse has two foci, and the sum of the distances fom any point on the ellipse to both foci is a constant. Impotant quantities ae: Semimajo axis a: This is half the distance acoss the longest diamete of the ellipse Eccenticity e: Vaies fom 0 to 1, with highe values indicating a moe elongated ellipse. Exteme cases: e = 0 cicle and e = 1 paabola (i.e., an obit that goes to infinity). Peihelion: The closest point to the Sun Aphelion: The fathest point fom the Sun The equation fo the Sunplanet ( sun ) distance as a function of a, e, and the angle η between the planet s pesent position and its peihelion (closest point to the Sun) is: sun = a(1 e2 ) 1+ ecosη 2. Planets move faste when they ae close to the Sun and slowe when they ae fathe away, so that an imaginay line connecting the Sun with a planet sweeps out equal aeas in equal times as it moves aound the Sun. A shot fom is equal aeas in equal times. Physically, this can be undestood in two ways: (i) Consevation of angula momentum of the planet obiting the Sun. Angula momentum is the coss poduct of distance and momentum, so as the distance deceases, angula velocity must incease. (ii) Consevation of enegy. As the planet falls towads the Sun it loses gavitational enegy, which is conveted into kinetic enegy, i.e., its velocity inceases. 8
2 3. The peiod of time it takes fo a planet to complete an obit, squaed, is popotional to the mean distance between the planet and the Sun, cubed. Mathematically: T = obital peiod a = semimajo axis k = 4π 2 /(GM)= constant,whee M = mass of Sun and G = gavitational constant Review of Newton s laws Newton s thee laws of motion combined with his law of gavity can explain planetay obits to fist ode. As a eminde: 1. Evey body continues in a state of est o unifom motion in a staight line unless acted on by a foce. 2. The acceleation of a body is popotional to the foce acting on it, and invesely popotional to the body's mass, i.e.,! F = m! a 3. To evey action (foce) thee is an equal and opposite eaction. (If one body exets a foce on anothe body, then the second body must also be exeting the same foce on the fist body, but in the opposite diection, e.g., fiing a gun, a ocket, mutual gavitational attaction, etc.) Newton's Law of Gavity! F 12 = Gm m 1 2! 2 e 12 G = Univesal Gavitational Constant = 6.674x1011 m 3 /(kg s 2 ) 12 = sepaation of the objects centes of masses Obits fo abitay masses Keple s laws apply when M >> m, i.e., one body is much moe massive than the othe, so that the moe massive body can be consideed to be fixed. Fo geneal masses m 1 and m 2 at!! vecto positions 1 and 2, both masses obit thei common cente of gavity (baycente). Actually, even fo staplanet systems, the sta moves a bit, exhibiting some wobble, which is how planets aound othe stas can be detected! Newton s laws can be used to deive the 2body dynamics: whee: combined with the gavity law gives: µ d 2! dt 2 = GµM' 2! e is the educed mass M'= m 1 + m! 2 =! 1! 2 is the total mass is the sepaation vecto of the two bodies 9
3 Hence, the elative motion is equivalent to that of a paticle of mass µ obiting a fixed cental mass M. Keple s laws can be genealized fo these abitay masses 1.) The 2 bodies move along elliptical paths, with one focus of each ellipse located at the!!! cente of mass cm = ( m m 2 1 ) / M' 2.) Equal aeas in equal times applies to  line connecting each body to cente of mass  line connecting the two bodies 4π 2 a 3 3.) The obital peiod is given by P 2 = G m 1 + m 2 ( ) Obital velocity, enegy and escape velocity Fo a cicula obit, the gavitational acceleation coesponds to the centipetal foce. Fom this, the obital velocity can be deived: Centipetal foce = gavitational foce To give some quantitative examples fo an object obiting the Eath: Just above Eath s suface, v = 8 km/s (=28,000 km/h) and the peiod P = 85 minutes. This is simila to the velocity and peiod of satellites in loweath obit. At the distance of the Moon (384,000 km) v = 3,723 km/h and P = 27 days (~1 month) At 35,680 km the peiod P = 24 h, i.e., as satellite obits in the same time as the Eath otates and stays the same point above Eath s suface. This geosynchonous obit is commonly used fo communication and weathe satellites. The gavitational potential enegy is usually defined to be zeo when two objects ae at infinite sepaation, and becomes inceasingly negative as the two objects ae bought close togethe: Let s compae this to the kinetic enegy of an object of mass m in a cicula obit aound the object if mass M. Fom the above equation fo v we get: E K = 1 2 mv 2 = 1 2 GMm Hence fo a cicula obit: 2E K = E G 10
4 and the total obital enegy E total = E K + E G = 1 E = E 2 G K Fo a noncicula obit, these ules apply to the timeaveaged enegies. So, obital enegy is always negative, and lage (less negative) enegy coesponds to a highe obit, and lowe total enegy coesponds to a lowe obit. If the object is given a total enegy that is positive, it can escape fom the othe object, i.e., goes to infinite distance. We can analyze this in tems of escape velocity, i.e., the velocity needed fo an object to escape fom the gavitational field of anothe object. Thinking in tems of enegy, its kinetic enegy must be equal to (o geate than) its gavitational potential enegy. v esc = 2GM = 2v obit Fo Eath, this escape velocity is 11.2 km/s = 40,250 km/h Actual sola system obits Most planets in ou sola system obit close to the ecliptic plane, counteclockwise when viewed fom the noth. The exception is Mecuy, whose obit exhibits an inclination of 7. Most moons obit close to the equatoial plane of thei planet, which is often inclined to the ecliptic plane because the planet s otation axis is tilted. This is the case fo Eath. Anothe example is Satun, whose otation axis is tilted by 27. Most moons obit thei planets counteclockwise (as planets obit the Sun) but 4 of Jupite's and 1 each of Satun's and Neptune's moons move counteclockwise, which is called etogade obit. Comets obits have andom inclinations, and often lage eccenticities. The most famous example is Halley s comet, which passes (since at least 2000 yeas) evey 76 yeas though the inne sola system. Fom Keple s 3 d law we can calculate that a = 18 AU. Its peihelion is obseved to be 0.54 AU, which gives it an eccenticity (Keple s 1 st law) of Its aphelion (geatest distance fom the Sun) is 35.5 AU. 2.2 The theebody poblem Unlike the 2body system, the system of 3 bodies inteacting gavitationally does not have simple solutions and dynamics is often chaotic. Vaious simplifications can be made. If one mass is much smalle than the othe two (m 3 << m 1 o m 2 ) this is called the esticted 3body poblem. In this case, m 1 & m 2 obey Keple s laws, and m 3 is consideed as a massless test paticle that does not influence the othe two. The question to be investigated is then: Whee does this test paticle go? If the second most massive body is futhe assumed to be on a cicula obit aound the most massive body, this is known as the cicula esticted 3body poblem. If they ae all in the same plane, this is called the plana 3body poblem. We discuss hee the plana cicula esticted 3body poblem. 11
5 To summaize, we assume that a massive body (e.g., sta) is obited by a less massive body (e.g., planet) on a cicula obit, and the motion of a thid body of negligible mass (e.g., atificial satellite o asteoid) is consideed, in the otating fame of efeence Lagange equilibium points In the efeence fame that is otating with the two most massive bodies, thee ae 5 points whee a test paticle will not move if placed caefully. These ae known as the Lagange points, and ae maked L 1 L 5. L 1 L 3 ae on a line connecting the two most massive bodies, wheeas L 4 and L 5 ae in the same obit but 60 degees to each side of the planet. L 1 L 2 : Eithe side of the planet, at the point whee the gavitational influence of the planet is balanced by the gavitational influence of the sta. These ae UNSTABLE, in the sense that if the test paticle is given a small velocity petubation, it will continue to move away fom these points. L 3 is close to the obit of the planet but on the othe side of the sta. This position is also UNSTABLE. L 4 L 5 ae STABLE, in the sense that if the test paticle is given a small petubation in velocity o position, it will libate (oscillate) aound these points, in an tajectoy discussed in the next section Tadpole and hoseshoe obits The obits of test paticles aound the L 4 and L 5 points (emembe, these obits ae supeimposed on the main obit of eveything aound the cental mass) can be constained by consideing the total enegy of a paticle in the otating fame of efeence. The equation is not given hee, but contous of equal enegy ae plotted on the figue in the notes. The subobit of a test paticle aound these points depends on its enegy  thee cases ae maked on the diagam: T = tadpole obit: An elongated subobit aound one of the L 4 o L 5 points H = hoseshoe obit: An elongated subobit that includes both L 4 and L 5 points. This can be undestood as follows: The paticle stats on a lowe obit than the planet, hence has a highe obital velocity and catches up with the planet. As it appoaches the planet it gains enegy though gavitational attaction and moves into a highe obit than the planet. This gives it a lowe angula velocity than the planet so it goes backwads in its elative obit. As it once again appoaches the planet, gavitational attaction causes it to lose enegy, falling again into the lowe obit. P = passing obit: Hee, the test paticle is fa enough fom the planet that it can continue on its obit with a mino petubation as it fist appoaches and then moves away fom the planet. These obits might not be smooth, but can have oscillations supeimposed on them. Thee ae seveal examples of L 4 and L 5 inteactions and obits in the sola system: Two moons of Satun, Janus and Epimetheus, ae in hoseshoe obits due to thei mutual gavitational inteaction. Epimetheus obit is much moe hoseshoelike because it is less massive than Janus. Thee ae 6083 known asteoids nea the L 4, L 5 points of Jupite  the Tojan asteoids 12
6 Asteoid (5261) Eueka is nea the L 5 point of Mas. Thee othe matian Tojans ae known. Small moons in the Satun system ae at the L 4 & L 5 points of the majo moons Tethys and Dione. The L 4 and L 5 points ae good locations fo satellites (e.g. NASA s STEREOA and STEREOB satellites) The Hill Sphee (gavitational influence of a planet) How fa does the gavitational influence of a planet extend? The Hill Sphee gives the adius at which the gavitational foce of the planet on a test paticle is equal to the diffeence between the gavitational foce of the Sun on the paticle and the gavitational foce of the Sun on planet (this is what mattes in the otating fame of efeence). " m planet R H = a$ $ # 3 m sun + m planet ( ) The Hill adius extends to the L 1 and L 2 Lagange points. This concept is useful fo consideing what happens when e.g., a comet comes close to a planet. If it comes within the Hill sphee it may get captued into obit aound the planet. All moons obit a planet within its Hill sphee. % ' ' & 1/ Resonances Many sola system bodies exhibit an intege atio of thei obital paametes, e.g., Obital peiod Io:Euopa:Ganymede = 4:2:1 Mecuy otates aound its spin axis within 1.5 obits. These ae examples of esonances caused by tidal locking. Tides ae discussed in the next section but locking mechanisms ae beyond the scope of this couse. Mecuy s obit and otation illustate the lage diffeence that can exist between the sideeal day: otation peiod elative to distant stas sola day: peiod fo the Sun to move once to its stat location as obseved fom the suface of the planet. Mecuy s sideeal day is 2/3 of its obital peiod, but its sola day is 2 obits! This is because it is otating slowly. Fo a apidlyotating body like Eath, the diffeence is much smalle. Convesely, in the asteoid belt thee ae gaps coesponding to obits that have an intege atio with Jupite s obit (e.g., 3:1) these ae called the Kikwood gaps. These ae examples of esonances caused by the gavitational petubation of Jupite on asteoid obits. Fo example in a 3:1 obit an asteoid will expeience a gavitational tug fom Jupite in the same place evey 3 obits. Ove many obits, these epeated tugs combine to make the asteoid obit chaotic, i.e., unpedictable. Eventually the asteoid obit changes to one that is outside this esonance. Thus esonant obits ae not stable ove long time peiods. 13
7 2.4 Tides and thei effect on obits and otation ates The tidal inteaction of the Eath and Moon was intoduced in the Geophysics couse, and is eviewed hee because it is impotant fo many moons and planets in the sola system. The gavitational pull of the Moon and the centifugal foce cause 2 bulges on Eath, which ae most obvious in the oceans but also exist (with smalle amplitude) in the solid pat of the Eath. The bulges ae appoximately aligned with the Moon while the Eath tuns elative to them, so that in the Eath s otating fame of efeence the bulges appea to move aound the Eath, with 2 passing evey day. The Sun s tidal pull may add o subtact fom this, causing exta lage o exta small tides (sping and neap tides). Because the Eath is not pefectly elastic, the location of the tidal bulges slightly lags behind thei coect positions. This lag causes the Moon to exet a net toque on the Eath s tidal bulges, which causes the Eath s otation to slow down. Fo the same eason, the Eath exets a slight tug on the Moon, giving it moe enegy and putting it into a moe distant obit. Thus, the Eath s otation slows down while the Moon moves futhe away. Quantitatively, Eath s otation slows by 1 s/50,000 a. Histoical measuements of eclipses ove the last 2500 yeas show that Eath s otation ate has slowed by 2.4 ms/centuy. 360 Ma ago, one day had 22 hous, as confimed by the fossil ecod. The Moon is moving away at 3.7 cm/a. This evolution will stop when the Eath s otation ate is the same as the Moon s obital peiod, i.e., they become tidally locked. This has aleady happened to the Moon: it s otational peiod = its obital peiod, so the same side always faces the Eath. It has also happened to most othe majo moons in the sola system, and to both bodies in the Pluto Chaon system. 2.5 Dissipative foces and the obits of small bodies So fa we discussed gavitational inteactions. Fo small bodies, othe foces may also be impotant, as summaized hee: Micometesize dust: Radiation pessue (of sunlight) pushes it away fom the Sun. Remembe that light can be consideed to be photons, which have momentum. When photons hit a suface, they exet a foce on it. Centimetesized paticles: Paticles of this size absob sunlight on one side and eemit it in all diections. In the fame of efeence of the paticle, it emits the same fequency in all diections (its tempeatue is ~constant). Fom the pespective of the gain of dust obiting the Sun, the Sun's adiation appeas to be coming fom a slightly fowad diection. Theefoe the absoption of this adiation leads to a foce with a component against the diection of movement. This is known as PoyntingRobetson dag. Mete to kilometesized objects: Lagesized bodies exhibit uneven tempeatue distibutions on thei sufaces. The diffeent tempeatues on thei sufaces cause them to emit diffeent momentum photons in diffeent diections, which changes thei obits. This is known as the Yakovsky effect. Whethe the object is migating inwads o outwads depends on its otation ate and diection. 14
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