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Lecture 2 - Minimum mass model of solar nebula

Lecture 2 - Minimum mass model of solar nebula. Topics to be covered: Composition and condensation Surface density profile Minimum mass of solar nebula. Terrestrial planets’ composition.

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Lecture 2 - Minimum mass model of solar nebula

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  1. Lecture 2 - Minimum mass model of solar nebula • Topics to be covered: • Composition and condensation • Surface density profile • Minimum mass of solar nebula PY4A01 Solar System Science

  2. Terrestrial planets’ composition • Mercury has a very large iron core about 3,500 km in diameter that makes up 60% of its total mass, surrounded by a silicate layer ~700 km thick. Its core is probably partially molten. • Mars has a solid Fe and/or iron-sulfide core ~2,600-4,000 km in diameter, surrounded by a silicate mantle and rocky crust that is probably several hundred km thick. • Venus' interior is like the Earth's, except its Fe-Ni core probably makes up a smaller percentage of its interior. PY4A01 Solar System Science

  3. Jovian planets’ interiors • Jupiter's H/He atmosphere is ~1,000 km thick and merges smoothly with the layer of liquid molecular H, which is ~20,000-21,000 km thick. Pressure near center is sufficient to create a liquid metallic H layer ~37,000-38,000 km thick. Probably has silicate/ice core twice diameter of Earth with ~14 times Earth's mass. • Saturn is smaller version of Jupiter: silicate core ~26,000 km in diameter, ice layer about 3500 km thick, beneath a ~12,000 km thick layer of liquid metallic H. Then liquid molecular H layer around 28,000 kilometers thick, and atmosphere about 2,000 km thick. • Compression on Uranus/Neptune probably not enough to liquefy H. Uranus/Neptune have silicate cores ~8,000-8,500 km in diameter surrounded by a slushy mantle of water mixed with ammonia and methane ~7,000-8,000 kilometers thick. At top is a 9000 -10000 km thick atmosphere of H and He. PY4A01 Solar System Science

  4. Minimum mass of the solar nebula • Can make approximation of minimum amount of solar nebula material that must have been present to form planets. Know: • Current masses, composition, location and radii of the planets. • Cosmic elemental abundances. • Condensation temperatures of material. • Given % of material that condenses, can calculate minimum mass of original nebula from which the planets formed. • Steps 1-8: metals & rock, steps 9-13: ices PY4A01 Solar System Science

  5. Nebula composition • Assume solar/cosmic abundances: PY4A01 Solar System Science

  6. Minimum mass for terrestrial planets • Mercury:~5.43 gcm-3 => complete condensation of Fe (~0.285% Mnebula). 0.285% Mnebula = 100 % Mmercury => Mnebula = (100/ 0.285) Mmercury = 350 Mmercury • Venus: ~5.24 g cm-3 => condensation from Fe and silicates (~0.37% Mnebula). =>(100% / 0.37% ) Mvenus = 270 Mvenus • Earth/Mars:0.43% of material condensed at cooler temperatures. => (100% / 0.43% ) Mearth = 235 Mearth • Asteroids: Cooler temperatures produce more condensation ~ 0.5 %. => (100% / 0.5%) = 200 Masteroids PY4A01 Solar System Science

  7. Minimum mass for terrestrial planets • What is the minimum mass required to make the Terrestrial planets? • Total of the 4th column is 29881x1026g. This is the minimum mass required to form the Terrestrial planets =>2.9881x1030g ~ 500 Mearth. PY4A01 Solar System Science

  8. Minimum mass for jovian planets and pluto • Jupiter: Almost nebula composition due to gas capture ~20%. => Mnebula =100 / 20 Mjupiter ~ 5 Mjupiteris minimum mass required. • Saturn: Cooler than Jupiter, with slightly different composition ~12.5%. => Mnebula = 100/12.5 Msaturn ~ 8Msaturn • Uranus: Less gas capture ~6.7% condensed to form planet. => Mnebula = 100/6.7 Muranus = 15 Muranus • Neptune: ~5% of solar nebula material condensed to form planet. => Mnebula = 100/5 Mneptune = 20 Mneptune • Pluto: Main fraction due to ices ~1.4 % => Mnebula = 100/0.14 Mpluto = 70 Mpluto PY4A01 Solar System Science

  9. Minimum mass for jovian planets • What is the minimum mass required to make the Jovian planets? • Total mass is therefore = 184450 x 1026g = 3085 Mearth. • This is minimum solar nebula mass required to make the Jovian planets. PY4A01 Solar System Science

  10. Minimum nebula mass • The minimum mass required to condense the nine planets is therefore: Planet M (x Mearth) Terrestrial 500 Jovian 3085 Pluto 0.119 3585 Mearth • This is the minimum mass required to produce the planets. • As Msun ~ 2 x 1033 g, the mass required to make the planets is therefore ~0.01 Msun. • Disk contained 1/100 of the solar mass. PY4A01 Solar System Science

  11. Nebular surface density profile • To make a more precise estimate, distribute min mass requirements over series of annuli, centred on each planet. • Choose boundaries of annuli to be halfway between the orbits of each planet. i.e., Mercury @ 0.38 AU and Venus @ 0.72 AU = > (0.72-0.38)/2 = 0.17 AU. • We therefore estimate that Mercury was formed from material within an annulus of 0.38±0.17 AU => 0.33 - 0.83 x 1013cm. • The surface density of an annulus,  = mass / area, where area =  router2-  rinner2 =  [(0.83 x 1013)2 - (0.33 x 1013)2] = 1.82 x 1026cm2 • Surface density of disk near Mercury is therefore: 1160x1026 / 1.82 x 1026 = 637 g cm-2 0.33x1013cm 0.83x1013cm PY4A01 Solar System Science

  12. Nebular surface density profile • For Venus at 0.72 AU, Mercury is at 0.38 AU and Earth is at 1 AU => Venus’ annulus extends from (0.72 - 0.38)/2 = 0.17 to (1 - 0.72)/2 = 0.14 • The material that formed Venus was located between 0.72 - 0.17 AU and 0.72 + 0.14 or 0.55-0.86 AU. This is 0.83-1.29 x 1013cm. • Area is then =  router2-  inner2= 3.06 x 1026 cm2. => = 13150 x 1026 / 3.06 x1 1026 = 4300 g cm-2. • This is the approximate surface density of the disk where Venus formed. • For Jupiter at 5.2 AU, the Asteroids are at 3 AU and Saturn is at 9.6 AU. The annulus therefore ranges from 4 - 7.2 or 6 - 11 x 1013cm. • As the area = 267 x 1013cm2 =>  = 95200 x 1026 / 267 x 1026 = 356 g cm-2 PY4A01 Solar System Science

  13. Minimum mass and density PY4A01 Solar System Science

  14. Surface density of solar nebula • Surface density of the drops off as:  (r) = 0 r - • 1 <  < 2, 0 ~ 3,300 g cm-3. • Local deficit of mass in asteroid belt. Mars is also somewhat deficient in mass. • Inside Mercury’s orbit, nebula material probably cleared out by falling in on Sun or blown out. • Outer edge may be due to a finite scale size of the original nebular condensation. PY4A01 Solar System Science

  15. Surface density of solar nebula • Hayashi et al. (1981) widely used:  (r) = 1700 (r / 1AU)-3/2 g cm-2 • Weidenschilling (1977) produced figure at right which shows similar trend. • Mars and asteroids appears to be under-dense. PY4A01 Solar System Science

  16. Surface density of solar nebula • Desch (2007) - disk much denser. • Disk much more massive: • 0.092 M in 1-30AU vs 0.011 M • Density falls steeply (as r-2.2) but very smoothly and monotonically. Matches to < 10%. • Nepture and Uranus must be switched in position => planetary migration? PY4A01 Solar System Science

  17. Minimum mass estimate • Can also estimate minimum mass from  : where RSis the radius of the Sun and RFis the max distance of Pluto. • Assume that ( r ) = 3300 ( r / RE )-2, where RE = 1 AU. Therefore, • Setting RE = 1.49 x 1013 cm, and RS = 6.96 x 1010 cm, and RF = 39 AU = > M 0.02 MSun • Ie approximately a factor of two of previous estimate. PY4A01 Solar System Science

  18. PY4A01 Solar System Science

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