Lectures 10-11: Planetary interirs Tpics t be cvered: Heat f frmatin Chemical differentiatin Natural radiactivity Planet cling Surface f Venus
Summary f planetary interirs Make-up f planetary interirs is dminated by physics f materials under high temperatures and pressures. Starting with cld, lw pressure regins, rcky materials are slids. As ne ges deeper int a planet, temperature and pressures rise. Slids becme semi-slid, plastic-like materials. With higher temperatures and pressures, semi-slids becme liquids. With even higher temperatures and pressures liquid r mlten rcky materials underg a phase change and becme slids again. This is why the very inner cres f the Earth and Venus are slid, surrunded by liquid uter cres.
Summary f terrestrial interirs Mercury has large irn cre abut 3500 km in diameter which is ~60% f its ttal mass, surrunded by a silicate layer ~700 km thick. Its cre is prbably partially mlten. Mars has a slid Fe and/r irn-sulfide cre ~2600-4000 km in diameter, surrunded by a silicate mantle and rcky crust that is prbably several hundred km thick. Venus' interir is like the Earth's, except its Fe-Ni cre prbably makes up a smaller percentage f its interir.
Summary f jvian interirs Jupiter's H/He atmsphere is ~1,000 km thick and merges smthly with the layer f liquid mlecular H, which is ~20,000-21,000 km thick. Pressure near center is sufficient t create a liquid metallic H layer ~37,000-38,000 km thick. Prbably has silicate/ice cre twice diameter f Earth with ~14 times Earth's mass. Saturn is smaller versin f Jupiter: silicate cre ~26000 km in diameter, ice layer abut 3500 km thick, beneath a ~12,000 km thick layer f liquid metallic H. Then liquid mlecular H layer arund 28,000 kilmeters thick, and atmsphere abut 2000 km thick. Cmpressin n Uranus/Neptune prbably nt enugh t liquefy H. Uranus/Neptune have silicate cres ~8000-8500 km in diameter surrunded by a slushy mantle f water mixed with ammnia and methane ~7000-8000 kilmeters thick. At tp is a 9000-10000 km thick atmsphere f H and He.
Heat f frmatin Initial planet internal temperature is due t accretin. Cnservatin f energy tells us: Ptential energy f material is cnverted int kinetic energy f mtin. R Upn hitting the planet, the kinetic energy f mtin is cnverted int internal heat energy f the planet - hence initial ht phase expected. Tw hemispheres with centers separated by distance R
Heat f frmatin Cnsider planet made f tw hemispheres f radius R and mass M/2. The ptential energy released, ΔU, by expanding these tw halves t infinity will be ΔU G ( M 2 ) 2 R R Cnservatin f energy says that the same amunt f energy will be liberated in the reverse prcess => E heat = ΔU
Heat f frmatin Amunt f energy (du) gained by additin f dm t bdy f mass M: du = GM dm r If bdy is spherical, M = 4/3 πρr 3 Additin f mass dm => dm = 4 πρ r 2 dr Substituting fr M and dm int equatin fr du and integrating gives: U = $ 3G& 4 % 3 πρ ' ) ( This is ttal energy ging int bdy frm accretin. R 0 = 3 5 G $ & 4 % 3 πρ ' ) ( = 3 GM 2 5 R 2 2 R 5 r 4 dr
Heat f frmatin Must cnsider specific heat capacity C P f a planet. Defined as amunt f heat energy required t raise unit mass f material by 1K. Rck: C P = 1200 J / kg / K Nickel: C P = 460 J / kg / K Ice: C P = 2100 J / kg / K S, change ΔT in temperature f a mass M due t E heat is: E heat = C P MΔT Equating this t energy due t accretin: C P MΔT = 3 GM 2 5 R Rearranging gives ΔT = 3 5 GM C p R Eqn (*) This is the maximum temperature f a planetary interir that results frm accretin.
Cnsequences Previus can be written ΔT = 4Gπρ R 2 p = 6.28 10 10 2 R p 5C P ΔT fr the Earth is large (40,000 K) and much greater than the melting temperature f rck (~1200 K). Setting ΔT ~ 1200 K (melting temperature f rck) => R~1000 km => Objects bigger than large asterids melt during accretin => Earth was a mlten blb f rck
Heat f frmatin Maximum temperatures attainable by accretin frm infinity: Object Radius (km) Mass (kg) ΔT (K) Earth 6,378 6 x 10 24 30,000 Venus 6,062 5 x 10 24? Mars 3,397 6 x 10 23 6,000 Did this energy dissipate at abut the same rate as it was generated r did it build up quicker than it culd be dissipated? Fr Earth think still sme remnant heat f frmatin. Fr smaller planets, heat f frmatin may have been dissipated as quickly as planet frmed.
Heat f frmatin Frm Hanks & Andersn (1969). http://www.gps.caltech.edu/uplads/file/peple/dla/dlapepi69.pdf
Temperature f frmatin Simulatin f impact between 80% M E prt-earth and 10% M E bject (Canup, R., Icarus, 2004) www.bulder.swri.edu/~rbin/
Cre pressures The central pressure is f rder the gravitatinal frce between the tw hemispheres divided by their area f interactin: P C = F grav / π R 2 Therefre, P C = (G/π) M 2 / R 4 This can be written in terms f the bulk density as P C = 1.4 x 10-10 ρ 2 R 2 (Pascals) Fr Jupiter, <ρ> = 1,300 kg/m 3 and R P = 7 x 10 7 m => P C (Jupiter) = 1.2 x 10 12 Pa = 12 Mega atmspheres
Cre pressures Jupiter and Saturn are principally cmpsed f hydrgen. At lw temperatures and pressure, H is an insulatr in the frm f the strngly bund diatmic mlecule H 2. At depths f a few thusand kilmeters belw the upper clud deck pressure becmes s high that the H 2 becmes dissciated and underges a phase transitin frm the gaseus t a liquid state. Fr P > 3 millin atmspheres, atms are inised int freely mving prtns and electrns. Phase is knwn as liquid metallic hydrgen (LMH). LMH is highly cnducting (the electrns are highly mbile) and this results in the generatin f a strng magnetic field.
Chemical differentiatin/fractinatin After planet frmatin, planets were hmgeneus. Fr bdies with diameters >few km, internal temperatures were large enugh t cause partial r ttal melting f the interirs => allwed materials t separate accrding t density. Undifferentiated Heavy materials are thse that bnd t irn, frming irn-bearing minerals (siderphiles, i.e., "irn-lvers ). Siderphiles sink t center f planet, frming cre. Light materials are thse the bnd t silicates; called the lithphiles. Rise t upper layers f planet. Separatin f materials accrding t density allws a cmplicated layered structure t frm inside a planet. Differentiated
Internal Heating frm Cre Separatin Differentiatin can be large heat surce sn after frmatin. Will ccur when temperature is large enugh fr metallic irn can sink t cre. What is gravitatinal energy released by this prcess? Cnsider tw-cmpnents, cre (r < r c ) and mantle (r > r c ), where r c is cre radius. Fr r < r c => M r,i = 4 / 3πr 3 ρ c Fr uter regins, r > r c => M r, = 4 / 3πr c 3 ρ c + 4 / 3π(r 3 r c 3 )ρ m = 4π 3 (ρ c ρ m )r 3 c + ρ m r 3
Internal Heating frm Cre Separatin Energy released is difference between gravitatinal energy f hmgenus bdy and bdy after differentiatin: where a = r c / R. r E g = G M r,i 4πr2 c ρ c R 0 dr +G M r, 4πr2 ρ m r r c dr r = 16π 2 GR 5 15 [ ρ 2 c a 5 + ρ 2 m (1 a 5 )+(5/2)ρ m (ρ c ρ m )a 3 (1 a 2 )] By cmparisn with E g = 3GM 2 / 5R, differentiatin releases abut ne-tenth as much energy as was riginally released frm accretin. See Landstreet Physical Prcesses in the Slar System: An intrductin t the physics f asterids, cmets, mns and planets http://www.astr.uw.ca/~jlandstr/planets/text/bk_asterids.ps
Chemical differentiatin/fractinatin Large planets (Earth/Venus) are mlten lng enugh fr a Fe and Ni cre t frm. Smaller planets (Mars) cl faster and slidify befre heavier elements sink t the cre => elements like Fe are ver abundant in the sil, giving Mars its red clr. Large planets cl slwer, and have thinner crusts. High cling rates als determine the interir structure. Slw cling rates imply sme planets still have warm interirs => mre diversified structure (inner cre, uter cre, semi-slid mantle, etc.) Earth is differentiated r layered; highest density in center, lwer densities prgressively utward Crust - rcky uter layer, brittle (5-40 km) Mantle - slid rcky layer, dense, high pressure, flw Outer Cre - mlten Fe-rich Inner Cre - slid Fe, Ni
Differentiatin within Earth Cmpsed f layers: Cre: Earth central prtin. Thickness ~3,470 km and temperature ~6,000K. Pressure s high cre is slid. Crust Pasty Magna: Prtin belw crust. Divided int mantle (thickness f 1200km) and intermediary layer (called external nucleus, thickness f 1700 km). Crust r Lithsphere: Earth s external layer. Thickness f 60 km in the muntainus areas and 5 t 10 km in the ceanic basins.
Cnditins within Earth Density (kg/cm -3 ) Temperature ( C)
Heating the planets Three main surces f heat: 1. Heat f accretin: Generated when planets accreted frm planetesimals. Clliding planetesimals cnvert gravitatinal ptential energy t kinetic energy and then thermal energy. 2. Heat f differentiatin: Generated at time planets separated int cre-mantle-crust. As mre dense material sinks, ptential energy cnverted t kinetic and then thermal energy. 3. Heat frm radiactive decay: Radiactive nuclei underg natural decay. Resultant particles cllide with neighburing atms. Accretin and differentiatin depsited heat billins f years ag. Radiactive decay is still a surce f heat, but was strnger in the past.
Heating by radiactive decay A number f radiactive istpes ccur naturally in rck. As they decay they prduce a heating effect. decay by ejecting e -, e + and α-particles. Sme examples: 40 K -> 40 Ca + e - + n Decay cnstant: λ( 40 K) = 5.54 x10-10 yr -1 => t 1/2 = 0.693 / λ = 1.25 Gyr 235 U -> 207 Pb + 7α + 4e - Decay cnstant: 9.85 x10-10 yr -1 => t 1/2 = 0.6 Gyr If radiactive decay is a majr cntributr t heating f planets, elements must have half-lives f ~Gyr.
Heating by radiactive decay Frm Landstreet (page 167) See article n Earth s heat budget and geneutrins: http://www.nature.cm/nge/jurnal/v4/n9/pdf/nge1240.pdf
Heating by radiactive decay If radiactive decay generates heat at a rate Q (J/kg/s), then the ttal amunt f energy generated per secnd in a unifrm spherical planet is: E RD = Q M planet = Q (4/3 π R 3 ) ρ Typical values fr Q are: Q ( 40 K) = 3.7 x 10-11 J/kg/s Q ( 235 U) = 4.6 x 10-12 Q ( 26 Al) = 2.5 x 10-8 Mst energetic radiactive decay prcess is due t decay f 26 Al Q ( 26 Al) = 676 Q ( 40 K) = 5435 Q ( 235 U) But, t 1/2 ( 26 Al) ~ 7 x 10 5 yr => nly imprtant sn after planet frmed.
Heating by radiactive decay Assume that 26 Al can supply energy fr t Al (yr). Temperature rise ΔT assciated with release f radiactive heat will be E RD = Q M t Al = M C P ΔT => ΔT = (Q / C P ) t Al Typically t Al ~ 10 6 years, and C P = 2,000 J/kg/s => ΔT ~ 400 K. Ttal heat liberated in first 1-2 Gyr wuld have been enugh t melt interirs f Earth and Venus. At present, ~50% f Earth s heat utput is thught t be due t radiactive heat prductin. Culd have been an rder f magnitude larger at time f planet frmatin.
Heating by radiactive decay
Radiactive heating f Earth since frmatin Frm Thermdynamics f the Earth and Planets by Albert Patin Duce
Cling f the planets Three main cling mechanisms: 1. Cnvectin (Q ~ dt/dr) in mantle carries ht rck t lithsphere where it slidifies. 2. Cnductin transfers heat frm the based f the lithsphere t the surface: Q = -k T ΔT erg cm -2 s -1 wherer k T is the thermal cnductivity. 3. Radiatin frm surface transfers heat t space: Q = Area x σt 4 ergs cm -2 s -1
Cling timescales Heat cntent f planet: E ~ M T ~ 4/3π R 3 T Assume planet cls by radiatin: de/dt = 4πR 2 σt 4 Therefre, cling time is: τ cling = E de /dt ~ R T 3 Implies: 1) Htter bdies cl faster that cler bdies and 2) Large bdies take lnger t cl. Explains why small bdies cled early, have large mantles, n magnetic fields, and crater-rich surfaces. Eg, R Mars /R Earth ~ 0.5 => Mars cled in half time f Earth. Des this explain Mars thick crust and lack f plate tectnics, vlcanic activity, magnetic field and thin secndary atmsphere?!
Cling timescales Cling rate smetimes given in terms f the surface-t-vlume rati: S/V ~ 1/τ cling High S/V fr Mercury => high cling rate => irn cre slidified and lwered internal temperature sufficiently t prevent vlcanic activity. Gelgical activity is directly related t interir temperature. See Intrductin t Planetary Science by G. Faure and T. M. Mensing.
Crust thickness Thickness f a planet's crust is determined by rate at which the planet cled. Fast cling rate (i.e. small planet) results thick crust. Fr majr terrestrial planets, crust thickness is prprtin t diameter. Cling rate is prprtinal t ttal mass f planet. Large planets cl slwer, have thinner crusts. High cling rates als determine the interir structure. Slw cling rates imply planets that still have warm interirs nw. A thicker crust als means less tectnic activity.
Terrestrial planet structure Small terrestrial planets: Interirs cled fast. Heavy cratering. N vlcanes. Mercury: unifrm cratering. Mn: highland heavily cratered, lava flws in maria. Medium terrestrial planets: Interir cled ff recently. Mderate cratering. Few extinct vlcanes. E.g., Mars. Large terrestrial planets: Interirs still cling tday. Light cratering. Many active vlcanes. Venus: vlcanes unifrmly spread. Earth: vlcanes at plate bundaries. Venus
Smaller terrestrial bjects The Mn Mars Radii < 4,000 km. N tectnic activity, heavily cratered. Mercury
An exceptin t the rule? Jupiter s mn I - evidence fr lava flws and vlcanic activity. Radius is 1,821 km => shuld therefre have cled ff early and slidified. Interir may be heated by tidal heating.
Tidal heating f I I is in synchrnus rbit arund Jupiter it keeps the same face tward Jupiter at all times (just like the Earth's Mn). Differential tidal frce f gravity stretches axis f planet alng planet-mn line. 2:1 resnance rbit with Eurpa => I make tw rbits fr every ne rbit that Eurpa Makes causes cnstant distrtin n I s shape ver its 1.8 days rbit f Jupiter.
Tidal heating f I Cnstant change in shape f I causes a large amunt f frictin in interir. This frictin generates a great deal f internal heat. This internal heat surce drives vlcanic activity n the surface f I. This heating is called Tidal Heating. T have Tidal Heating, need: A massive central planet (Tidal frces depend n mass) A mn rbiting clse t the massive planet (Tidal frces depend n distance). Anther mn in resnance with the inner mn. (Need eccentric rbit in rder t keep distance between inner mn and planet changing)
I's vlcan Tvashtar
Cling timescale fr Jvian planets Heating during accretin is likely t be the larges heat surce fr planets. The gain in energy due t accreting frm a distance r is ver a time dt is: " E acc = GM(r) % $ σ (T 4 (r) T 4 0 )'dt # r & If accretin is rapid, much f heat f accretin is stred inside planet: GM(r) C P MΔT r If planet cls by radiating energy, it luminsity is: L = 4πR 2 σ(t 2 e - T eq2 ) The rate f change f the mean internal temperature is therefre, dt dt = L C V M where C V is the specific heat at cnstant vlume (use C V as assume adiabatic).
Cling timescale fr Jvian planets The cling time can therefre be estimated via Δt ΔTMC V L Wrks well fr Jupiter and pssibly Saturn, where adiabatic heat transfer hlds. Fr Uranus and Neptune, equatin fr thermal evlutin can be written: Cf dt e dt = (T e 4 T eq 4 ) where C is a cnstant that characterises the thermal inertia f the interir, and f is the fractin f the internal heat reservir that gives rise t the bserved luminsity. Drp in temperature fr Uranus and Neptune ver age f Slar System is ~200 K, which is small cmpared t larger gas giants (see de Pater and Lissauer, Planetary Sciences, fr further details).
Energy budgets fr Jvian planets Observatins shw Jupiter, Saturn and Neptune radiate mre energy int space than they receive frm the Sun (fr sme reasn Uranus des nt) => must have an internal energy surce. Planet Rati radiated t slar absrbed Internal Pwer (Watts) Jupiter 1.67 ± 0.08 4 x 10 17 Saturn 1.79 ± 0.10 2 x 10 17 Uranus < 1.4 < 10 15 Neptune 2.7 ± 0.3 3 x 10 15 Since the Jvian planets are principally cmpsed f gas and ice, radiactive heating will nt be imprtant. Energy surce is gravitatinal energy release thrugh shrinking.
Energy generatin fr Jvian planets Cnsider a simple cre plus envelpe mdel f a Jupiterlike planet f ttal mass M P. If the radius f the envelpe changes frm R i t R f, cnservatin f energy requires that R i Envelpe (M env ) ΔK GM R cre f M env = GM M R cre i env where DK is the (kinetic) energy prduced by the cntractin f the envelpe (change in PE) Cre (M cre ) Therefre, 1 R f 1 R i = ΔK GM M cre env
Energy generatin fr Jvian planets Fr Jupiter: M cre = M env = M Jupiter / 2 = 9.5 x 10 26 kg ΔK = 4 x 10 17 Watts R i = 7.1 x 10 7 m (present radius) If we set R f = R i +ΔR, then t generate the bserved ΔK, Jupiter has t shrink by an amunt: ΔR = -1 mm / yr This rate f change in the radius amunts t 1 km in a millin years - a change that is far t small t measure. Hence the excess energy radiated by Jupiter int space can be easily accmmdated by small amunts f shrinkage in its gaseus envelpe.