Problem #1 [Sound Waves and Jeans Length]


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1 Roger Griffith Astro 161 hw. # 8 Proffesor ChungPei Ma Problem #1 [Sound Waves and Jeans Length] At typical sealevel conditions, the density of air is gcm 3 and the speed of sound is cm sec 1. Find (a) the jeans length and comment on how it compares with the thickness of the atmosphere and if you expect Jeans instability to occur; (b) the fractional change in frequency due to the selfgravity of the air, for a sound wave with wavelength 1 meter. the Jeans length is given by λ J = 2π k J where k J is the Jeans wave number which is given by 4πGρ k J = v 2 s where v 2 s is the characteristic sound speed, thus the Jeans length is πv λ J = 2 s = cm Gρ This means that Jeans instability will not occurr, due to the fact that the thickness oh the atmosphere is than the Jeans wavelength. (b). to find the fractional change in frequency we must use ω ω = ω ω J = v sk v s (k 2 kj 2) = 1 ω v s k which yields (k 2 k 2 J ) k ω ω ω ω = 1 λ ( (2π) 2 2π λ 2 4πGρ ) 1/2 v 2 s = 1 1 Gρ λ 2 πv 2 = s 1
2 Problem #2 [No More Jeans Swindle] The Jeans instability can be analyzed exactly, without invoking the Jeans swindle, in certain cylindrical rotating systems. Consider a homogeneus, selfgravitating fluid of density ρ, contained in an infinite cylinder of radius R. The cylinder walls and fluid rotate at uniform angular speed Ω = Ωz, where z lies along the axis of the cylinder. The Euler equation for this rotating system is v +( v ) v = 1 ρ P φ 2 Ω v+ω 2 (x x+y y) where the additional terms are the Coriolis and centrifugal forces. (a). Show that the gravitational force per unit mass inside the cylinder is φ = 2πGρ (x x+y y) We can solve this problem by using Gauss s law, which states where the M enc and the A are given by F g A = 4πGM enc this gives us M enc = πr 2 hρ A = 2πR h F g (2πR h) = 4πG(πR 2 hρ ) F g = 2πGρ R but we know that F = F ˆr = 2πρ GR ˆr but R ˆr = (x ˆx+yŷ), so we find F = φ = 2πGρ (x ˆx+yŷ) (b). Find the condition on Ω so that the fluid is in equilibrium with zero velocity and no pressure gradients. 2
3 The conditions needed for this problem are The Euler equation is applying these conditions we find thus, we find v = P = v +( v ) v = 1 ρ P φ 2 Ω v+ω 2 (x x+y y) φ = Ω 2 (x ˆx+yŷ) = 2πGρ (x ˆx+yŷ) Ω = 2πGρ (c). Let R so that the boundery condition due to the wall can be neglected. Find the dispersion relation for waves propogating parallel to the rotation axis z. Discuss if these waves are stable. we know that v 1 = e i(kz ωt) = (v x ˆx+v y ŷ+v z ẑ)e i(ks ωt) P 1 = v 2 s φ 1 = 2 φ 1 = 4πG We must use these realtionships to linearize the three fluid equation, the linearized equation are given as v 1 = 1 P1 ρ φ 1 2 Ω v 1 equation 1 = ρ ( v 1 ) equation 2 2 φ 1 = 4πG equation 3 We can take a time derivative of equation 2 to get 2 = ρ ( v 1 ) = ρ 1 ( P1 ρ φ 1 2 Ω v 1 ) which gives us 2 [ ] 1 2 = ρ 2 P 1 ρ 2 φ 1 2 (Ωv x ŷ Ωv y ˆx)e i(kz ωt) [ ] 1 = ρ 2 P 1 ρ 2 φ 1 = 2 P 1 + ρ 2 φ 1 3
4 since we know that we can just plug this in to find P 1 = v 2 s 2 φ 1 = 4πG since we also know that we find 2 = v2 s 2 + ρ 4πG = e i(kz ωt) therefore we find 2 = ω 2 e i(kz ωt) = ω 2 2 = ( k) 2 e i(kz ωt) = k 2 ω 2 = v 2 s k 2 + ρ 4πG thus we find the dispersion relationship to be ω 2 = v 2 sk 2 ρ 4πG (d). Find the dispersion relation for waves propogating perpendicular (you may pick x without loss og generality) to the rotation axis z. Discuss if these waves are stable. We will solve this problem the same way as part (c), we can begin with we need to solve for 2 = ρ ( v 1 ) = ρ 1 ( P1 ρ φ 1 2 Ω v 1 ) Ω v1 = v 1 Ω Ω v 1 = Ω v 1 so we find but we know that 2 = ρ ( v 1 ) = ρ 1 ( P1 ρ φ 1 2( Ω v 1 )) v 1 = 1 ρ P1 φ 1 2 Ω v 1 4
5 so v 1 ( = 2 Ω v 1 = 2 (Ωv x ŷ Ωv y ˆx) = 2Ω v x z ˆx v ( y z ŷ+ vx x v ) ) y ẑ y = 2Ω v x x ẑ = d dt v 1 we also know that and so therefore thus and as before ( = ρ vx v1 = ρ x + v y y + v ) z z = ρ v x 2Ω x = ρ 2Ω ( v 1 )z Ω ( v 1 ) = Ω( v 1 )z = 2Ω2 ρ 2 [ ( 1 2Ω 2 = ρ 2 P )] φ 1 2 ρ ( 2Ω ω 2 = 2 P 1 + ρ 2 2 ) φ 1 + 2ρ ρ ω 2 = v 2 s k 2 + ρ 4πG+4Ω 2 Thus we find the dispersion relationship to be ω 2 = v 2 s k 2 4πGρ 4Ω 2 ρ 5
6 Rastika s The Formation of Galaxy Structure and Evolution of Morphologies 1. Physics of galaxy formation. a. How do galaxies form from the primordal gas? b. Did most galaxies for around the same epoch? 2. Formation Theories a. Monolithic: Since stars with low metallicity had very low angular momentum L z, they suggested that the old stars were formed out of gas falling towards the center in radial orbits, collapsing quickly from a halo to a thin rotating disk plane enriched in heavy elements by star formation. b. Hierarchical: A system like our own galaxy is the result of the hierarchical assembly of dark halo building blocks. Accretion of baryoinic gas occurs later, in the assembled structure, to form the bulge, and progressivly the thin disk, which forms last. c. Secular: In secular evolution the bulge component is formed slowly from the disk through the bar interaction, and the disk can be replenished through continues external gas accretion. 3. Galaxies have different morphologies. a. Ellipticals: Most of the largest galaxies that we observe are elliptical galaxies, many elliptical galaxies are believed to form due to the interaction of galaxies, resulting a collision or merger. b. Spirals: Spiral galaxies consist of a rotating disk of stars, along with a central bulge of generally older stars. Extending outwards from the bulge are sometimes relative bright arms. There are many subclasses for each galaxy morphology. c. Lenticular galaxies: A lenticular galaxy is an intermediate form that has properties of both elliptical and spiral galaxies. Roger s 1. Observational results from the Hubble Space Telescope ACS Extended Groth Strips a. Multiwavelength images have been obtained from the HST in both the I band (F814) and V band (F66) as part of the AEGIS collaboration. b. We have used a paramteric technique to measure galaxy morphology parameters. 2. Measuring the Sersic index and the effective radius using Galfit. 3. Quantifying galaxy morphologies : comparing Galfit to the Gini coefficient(concentration) and M2(assymetry). 6
7 4. Corralating galaxy morphology to Sersic index. 5. Morphologies at different wavelengths and at differen lookback times 7
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