Tidal evolu+on. Tidal deforma+on Rota+onal deforma+on Tidal torques Orbit decay and spin down rates Tidal circulariza+on Hot Jupiters

Transcription

1 Tidal evolu+on Tidal deforma+on Rota+onal deforma+on Tidal torques Orbit decay and spin down rates Tidal circulariza+on Hot Jupiters

2 Tidal and rota+onal deforma+ons Give informa+on on the interior of a body, so are of par+cular interest Tidal forces can also be a source of heat and cause orbital evolu+on

3 Poten+als of non- spherical bodies External to the planet the gravita+onal poten+al is zero 2 Separate in r,φ,θ Can expand in spherical harmonics with general solu+on (r,,µ)= Given axisymmetry the solu+ons are only a func+on of μ=cos(θ) and are Legendre Polynomials, with n =0 (A n r n + B n r (n+1) )S n (µ, ) P 1 (µ) = µ P 2 (µ) = 1 2 (3µ2 1)

4 Poten+al External to a non- spherical body For a body with constant density γ body, mean radius C and that has an equatorial bulge R( ) =C(1 + 2 P 2 (cos( )) Has exterior gravita+onal poten+al (r, ) = 4 3 C3 G 1 r And internal poten+al (r, ) = 4 apple 3C 2 r 2 3 C3 G 2C C 2 r 3 2P 2 (cos ) r 2 C 3 2P 2 (cos )

5 Poten+al external to a non- spherical body Quadrupole component only 2 GMR2 r 3 2 P 2 (cos ) Form useful for considered effect of rota+on and +dal forces on shape and external poten+al

6 Rota+onal deforma+on Can define an effec+ve poten+al which includes a centripetal term FlaTening depends on ra+o of centrifugal to gravita+onal accelera+on Exterior to body cf = 2 r 2 3 [P 2(cos ) 1] q = 2 R 3 Gm p (r, )= Taking J 2 term only and assuming isopoten+al surface GM r 1 n J n factor of 3 from defini+on of P 2 R r n P n (cos ) Gm p gh + J 2 R + 2 R 2 3 f = 3 2 J 2 + q 2 =0 rewrite this in terms of difference between pole and equatorial radius, f and flatening q parameter

7 Rota+onal deforma+on It is also possible to relate J 2 to moments of iner+a J 2 C A Ma 2 Where C,A are moments of iner+a and a is a long axis of the body

8 Rota+onal deforma+on Again assume surface is a zero poten+al equilibrium state Balance J 2 term with centrifugal term to relate oblateness (f=δr/r) to moments of iner+a (J 2 ) and rota+on f = 3 2 J 2 + q 2 Observables from a distance: oblateness f, spin parameter, q Resul+ng constraint on moments of iner+a which gives informa+on on the density distribu+on in the body J 2 can also be measured from a flyby or by observing precession of an orbit

9 Tidal Deforma+on satellite m s planet m p semi- major axis a radius R

10 Tidal Deforma+on Tidal deforma+on by an external object In equilibrium body deforms into a shape along axis to perturber R( ) =C(1 + 2 P 2 (cos( )) Surface is approximately an equipoten+al surface That of body due to itself must balance that from external +de. Tidal poten+al V T Body assumed to be in hydrosta+c equilibrium Height x surface accelera+on is balanced by external +dal perturba+on gh V T h = C g = Gm p C 2 C = mean radius surface accelera+on

11 Tidal force due to an external body External body with mass m s, and distance to planet of a Via expansion of poten+al the +dal force from satellite onto planet is ( )= G m s a 3 R2 pp 2 (cos ) Where ψ is defined as along satellite planet axis If we assume constant density then this can be equated to poten+al term of a bulging body to find ε the size of the equatorial deforma+on from spherical More sophis+cated can treat core and ocean separately each with own density and boundary and using the same expressions for interior and exterior poten+al

12 Planet s poten+al equipoten+al surface h(ψ) and accelera+on gh( ) g = GM p Rp 2 accelera+on +mes height above mean surface is balanced by +dal poten+al at the surface = Gm s a 3 R2 pp 2 (cos ) for these to balance h(ψ) must be propor+onal to P 2 (cos ψ) 3 Rp m s h R p M p a The poten+al generated by the deformed planet, exterior to the planet V p GM p r 3 V p Gm s a RpP 2 2 (cos ) h R p 5 Rp Gm s P 2 (cos ) = k 2 a a Rp a Love number k 2 5 P 2 (cos ) Details about planet s response to the +dal field is incorporated into the unitless Love number

13 Love numbers Surface deforma+on Poten+al perturba+on h R = h V T (R) 2 g V 2 (r, ) = k 2 V T (R) R r 3 P 2 (cos ) h 2, k 2 are Love numbers They take into account structure and strength of body For uniform density bodies h 2 = 5/2 1+ µ µ 19µ 2 gr dimensional quan+ty ra+o of elas+c and gravita+onal forces at surface μ is rigidity k 2 = 3 5 h 2

14 Love numbers h 2 = 5/2, k 2 =3/2 for a uniform density fluid These values some+mes called the equilibrium +de values. Actual +des can be larger- not in equilibrium (sloshing) For s+ff bodies h 2,k 2 are inversely propor+onal to the rigidity μ and Love numbers are smaller Constraints can be made on the s+ffness of the center of the Earth based on +dal response

15 Tidal torque satellite m s lag angle δ planet m p semi- major axis a radius R

16 Tidal Torques Dissipa+on func+on Q = E/dE, energy divided by energy lost per cycle; Q is unitless For a driven damped harmonic oscillator the phase shii between response and driving frequency is related to Q: sin δ=- 1/Q Torque on a satellite depends on difference in angle between satellite planet line and deforma+on angle of satellite = m s and this is related to energy dissipa+on rate Q

17 Tidal Torques Torque on satellite is opposite to that on planet However rates of energy change are not the same (energy lost due to dissipa+on) Energy change for rota+on is ΓΩ (where Ω is rota+on rate and Γ is torque) Energy change on orbit is Γn where n is mean mo+on of orbit. Angular momentum is fixed (dl/dt =0)so we can relate change of spin to change of mean mo+on L = I + m pm s m p + m s a 2 n = 1 2I m p m s m p + m s naȧ

18 Tidal Torques E = 1 2 I 2 Gm p m s 2a Ė = I m p m s m p + m s n 2 aȧ m p m s m p + m s naȧ( n) Subbing in for dω/dt Ė = 1 2 As de/dt is always nega+ve the sign of (Ω- n) sets the sign of da/dt If satellite is outside synchronous orbit it moves outwards away from planet (e.g. the Moon) otherwise moves inwards (Phobos)

19 Orbital decay and spin down Deriva+ve of poten+al w.r.t ψ depends on P 2 (cos ) = 3 2 sin 2 Torque depends on this deriva+ve, the angle itself depends on the dissipa+on rate Q ȧ = sign( p n) 3k 2 m s Q p where k 2 is a Love number used to incorporate unknowns about internal structure of planet Compu+ng the spin down rate (α p describes moment of iner+a of planet I=α p m p R 2 p ) p = sign( p n) m p C p a 5 na 3k 2 2 p Q p m 2 s m p (m p + m s ) C p = R p C p a 3 n 2

20 Orbital decay ȧ = sign( p n) 3k 2 Q p m s m p C p a 5 na Tidal +mescales for decay tend to be strongly dependent on distance. Quadrupolar force drops quickly with radius. Strong power of radius is also true for gravita+onal wave decay +mescale

21 Mignard A parameter Satellite +de on planet Planet +de on satellite How to es+mate which one is more important if both are rota+ng? A mp m s 2 Rs R p 5 k 2s k 2p Q p Q s A way to judge importance of +des dissipated in planet vs those dissipated in satellite Spin down leads to synchronous rota+on When both bodies are +dally locked +dal dissipa+on ceases

22 Eccentricity damping or Tidal circulariza+on Neglect spin angular momentum, only consider orbital angular momentum that is now conserved conserved e 2 =1+ 2EL2 orb G 2 (m p + m s ) (m p m s ) 3 ė = Ė 2eE (1 e2 ) Ė 2eE As de/dt must be nega+ve then so must de/dt Eccentricity is damped This rela+on depends on (de/dt)/e so is directly dependent on Q as were our previously es+mated +dal decay rates (Eccentricity can increase depending on Mignard A parameter). Moon s eccentricity is currently increasing.

23 Tidal circulariza+on (amplitude or radial +de) weaker +de stronger +de Distance between planet and satellite varies with +me Tidal force on satellite varies with +me Amplitude varia+on of body response has a lag energy dissipa+on

24 Tidal circulariza+on (libra+on +de) In an ellip+cal orbit, the angular rota+on rate of the satellite is not constant. With synchronous rota+on there are varia+ons in the +lt angle w.r.t to the vector between the bodies Lag energy dissipa+on

25 Tidal circulariza+on Consider the gravita+onal poten+al on the surface of the satellite as a func+on of +me The +me variable components of the poten+al from the planet due to planet to first order in e V s = Gm pcs 2 3e cos ntp2 a 3 (cos )+3esin 2 sin 2 sin nt radial +de libra+on +de β angle between posi+on point in satellite and the line joining the satellite and guiding center of orbit θ azimuthal angle from point in satellite to satellite/planet plane φ azimuthal angle in satellite/planet plane

26 Response of satellite For a s+ff body: The satellite response depends on the surface accelera+on, can be described in terms of a rigidity in units of g Vt ~μ g h h m pcs 4 e a 3 µ Total energy is force x distance (actually stress x strain integrated over the body) F Gm pc s a 3 em s W Fh Gm2 p a Cs 5 e 2 a µ This is energy involved in +de Using this energy and dissipa+on factor Q we can es+mate de/dt Ė E n Q s

27 Eccentricity damping or Tidal circulariza+on Eccentricity decay rate for a s+ff satellite e = 4 63 m s m p where μ is a rigidity. Sum of libra+on and radial +des. A high power of radius. a C s 5 µ s Q s n Similar +mescale for fluid bodies but replace rigidity with the Love number. Decay +mescale rapidly become long for extra solar planets outside 0.1AU

28 Tidal forces and resonance If +dal evolu+on of two bodies causes orbits to approach, then capture into resonance is possible Once captured into resonance, eccentrici+es can increase Hamiltonian is +me variable, however in the adiaba+c limit volume in phase space is conserved. Assuming captured into a fixed point in Hamiltonian, the system will remain near a fixed point as the system driis. This causes the eccentricity to increase Tidal forces also damp eccentricity Either the system eventually falls out of resonance or reaches a steady state where eccentricity damping via +de is balanced with the increase due to the resonance

29 Time delay vs phase delay Prescrip+ons for +dal evolu+on Darwin- Mignard +des: constant +me delay Darwin- Kaula- Goldreich: constant phase delay phase angle spinning object synchronously locked If Δt is fixed, then phase angle and Q (dissipa+on) changes with n (frequency) Lunar studies suggest that dissipa+on is a func+on of frequency (Efroimsky & Williams, recent review) Tidal evolu+on taking into account normal modes: see Jennifer A. Meyer s recent work

30 Tides expanded in eccentricity For two rota+ng bodies, as a func+on of both obliqui+es and +des on both bodies

31 Tidal hea+ng of Io Peale et al es+mated that if Qs~100 as is es+mated for other satellites that the +dal hea+ng rate of Io implied that it s interior could be molten. This could weaken the rigidity and so lead to a runaway mel+ng events. Io could be the most intensely heated terrestrial body in the solar system Morabita et al showed Voyager I images of Io displaying prominent volcanic plumes. Voyager I observed 9 volcanic errup+ons. Above is a Gallileo image

32 Tidal evolu+on The Moon is moving away from Earth. The Moon/Earth system may have crossed or passed resonances leading to hea+ng of the Moon, its current inclina+on and affected its eccentricity (Touma et al.) Satellite systems may lock in orbital resonances Es+ma+ng Q is a notoriously difficult problem as currents and shallow seas may be important Q may not be geologically constant for terrestrial bodies (in fact: can t be for Earth/moon system) Evolu+on of satellite systems could give constraints on Q (work backwards)

33 Consequences of +dal evolu+on Inner body experiences stronger +des usually More massive body experiences stronger +dal evolu+on Convergent +dal evolu+on (Io nearer and more massive than Europa) Inner body can be pulled out of resonance w. eccentricity damping (Papaloizou, Lithwick & Wu, Batygin & Morbidelli) possibly accoun+ng for Kepler systems just out of resonance

34 Sta+c vs Dynamic +des We have up to this +me considered very slow +dal effects Internal energy of a body Gm 2 /R Grazing rota+on speed is of order the low quantum number internal mode frequencies (Gm/R 3 ) 1/2 During close approaches, internal oscilla+on modes can be excited Energy and angular momentum transfer during a pericenter passage depends on coupling to these modes (e.g. Press & Teukolsky 1997) Tidal evolu+on of eccentric planets or binary stars Tidal capture of two stars on hyperbolic orbits

35 Energy Deposi+on during an encounter following Press & Teukolsky 1977 Ė = Z (r,t)= Z E = dv v r GM r R(t) dtė dissipa+on rate depends on mo+ons in the perturbed body perturba+on poten+al velocity w.r.t to Lagrangian fluid mo+ons total energy exchange Describe both poten+al and displacements in terms of Fourier components Describe displacements in terms of a sum of normal modes Because normal modes are orthogonal the integral can be done in terms of a sum over normal modes

36 Energy Deposi+on due to a close encounter Poten+al perturba+on described in terms of a sum over normal modes r = X A n (!) n E / X A(!) 2 dimensionless expression internal binding energy of star E = GM 2 R m M energy dissipated depends on overlap integrals of +dal perturba+on with normal modes r s M d 3 d=pericenter M + m R 3 strong dependence on distance of pericenter to star 2 X 2l+2 R T l ( ) d mul+pole expansion l to es+mate dissipa+on and torque, you need to sum over modes of the star/planet, oien only a few modes are important

37 Capture and circulariza+on Previous assumed no rela+ve rota+on. This can be taken into account (e.g., Invanov & Papaloizou 04, 07 ) quasi- synchronous state that where rota+on of body is equivalent to angular rota+on rate a pericenter Excited modes may not be damped before next pericenter passage leading to chao+c varia+ons in eccentricity (work by R. Mardling)

38 Tidal predic+ons taking into account normal modes of a planet Recent work by Jennifer Meyer on this

39 Hot Jupiters Cri+cal radius for +dal circulariza+on of order 0.1 AU Rapid drop in mean eccentricity with semi- major axis. Can be used to place a limit on Q and rigidity of these planets using the circulariza+on +mescale and ages of systems Large eccentricity distribu+on just exterior to this cutoff semi- major axis Large sizes of planets found via transit surveys a challenge to explain.

40 Possible Explana+ons for large hot Jupiter radii They are young and s+ll cooling off They completely lack cores? They are +dally heated via driving waves at core boundary rather than just surface (Ogilvie, Lin, Goodman, Lackner) Gravity waves transfer energy downwards Obliquity +des. Persistent misalignment of spin and orbital angular momentum due to precessional resonances (Cassini states) (A possibility for accoun+ng for oceans on Europa?) Evapora+on of He (Hanson & Barman) Strong fields limit loss of charged Hydrogen but allow loss of neutral He leading to a decrease in mean molecular weight Ohmic dissipa+on (e.g., Batygin) Kozai resonance (e.g., Naoz)

41 Cooling and Day/Night temperatures Radia+on cooling +mescale sets temperature difference If thermal contrast too large then large winds are driven day to night (advec+on) Temperature contrast set by ra+o of cooling to advec+on +mescales (Heng)

42 Quadrupole moment of non- round bodies For a body that is uneven or lopsided like the moon consider the ellipsoid of iner+a (moment of iner+al tensor diagonalized) Euler s equa+on, in frame of rota+ng body = L + L = I + L Set this torque to be equal to that exerted +dally from an exterior planet

43 Spin orbit coupling spin of satellite w.r.t to its orbit about the planet Fixed reference frame taken to be that of the satellite s pericenter = f spinning A B moments of iner+a A,B,C are from moment of iner+a tensor Introduce a new angle related to C 3 2 (B A)Gm p r 3 sin 2 =0 mean anomaly of planet A resonant angle = pm p = i/j = p is integer ra+o

44 Spin orbit resonance n2 B C A a 3 sin(2 +2pM 2f) r Now can be expanded in terms of eccentricity e of orbit using standard expansions If near a resonance then the angle γ is slowly varying and one can average over other angles Finding an equa+on that is that of a pendulum sin 2

45 Spin Orbit Resonance

46 Spin Orbit Resonance and Dynamics Tidal force can be expanded in Fourier components Contains high frequencies if orbit is not circular Eccentric orbit leads to oscilla+ons in +dal force which can trap a spinning non- spherical body in resonance If body is sufficient lopsided then mo+on can be chao+c

47 M+D Chap 4, 5 Reading