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Problem #1 [Sound Waves and Jeans Length]

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Roger Griffith Astro 161 hw. # 8

Proffesor Chung-Pei Ma

Problem #1 [Sound Waves and Jeans Length]

At typical sea-level conditions, the density of air is 1.23×10−3gcm−3 and the speed of sound is 3.4×104cm 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 self-gravity of the air, for a sound wave with wavelength 1 meter.

the Jeans length is given by

λJ=

kJ

where kJ is the Jeans wave number which is given by

kJ=

s

Gρ0 v2

s

where v2s is the characteristic sound speed, thus the Jeans length is λJ=

s

πv2

s

Gρ0 = 6.65×109cm

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−ωJ ω = vskvs q (k2k2 J) vsk =1− q (k2k2 J) k which yields ∆ω ω = 1− λ 2π (2π)2 λ2 − 4πGρ0 v2 s 1/2 ∆ω ω = 1− s 1−Gρ0λ 2 πv2 s = 1−p1−2.26×10−16 0

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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, self-gravitating fluid of densityρ0, con-tained in an infinite cylinder of radius R0. 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

∂t + (~v·~∇)~v=− 1

ρ~∇P−~∇φ−2~Ω×~v+Ω2(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

−~∇φ0=−2πGρ0(x~x+y~y)

We can solve this problem by using Gauss’s law, which states Fg·A=4πGMenc

where the Mencand the A are given by

Menc = πR20hρ0 A = 2πR0h this gives us

Fg(2πR0h) = 4πG(πR20hρ0) |Fg| = 2πGρ0R0 but we know that

~

F =−|F|ˆr=−2πρ0GR0ˆr but R0ˆr= (x ˆx+y ˆy), so we find

~

F=−∇φ0= −2πGρ0(x ˆx+y ˆy)

(b). Find the condition onΩso that the fluid is in equilibrium with zero velocity and no pressure gradients.

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The conditions needed for this problem are

~v0=0 ~∇P=0 The Euler equation is

∂~v

∂t + (~v·~∇)~v=− 1

ρ~∇P−~∇φ−2~Ω×~v+Ω2(x~x+y~y) applying these conditions we find

~∇φ=2(x ˆx+y ˆy) =2πGρ0(x ˆx+y ˆy) thus, we find

Ω=p2πGρ0

(c). Let R0 →∞ 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

ρ1 = ρ1ei(kz−ωt)

~v1 = (vxxˆ+vyyˆ+vzˆz)ei(ks−ωt)

P1 = v2sρ1

φ1 = ∇2φ1=4πGρ1

We must use these realtionships to linearize the three fluid equation, the linearized equation are given as ∂~v1 ∂t = 1 ρ0~∇P1−~∇φ1−2~Ω×~v1 equation 1 ∂ρ1 ∂t = −ρ0(~∇·~v1) equation 2 ~2φ1 = 4πGρ1 equation 3 We can take a time derivative of equation 2 to get

∂2ρ1 ∂t2 =−ρ0 ∂(~∇·~v1) ∂t =−ρ0~∇·( 1 ρ0~∇P1−~∇φ1−2~Ω×~v1) which gives us ∂2ρ1 ∂t2 = −ρ0 1 ρ0~∇2P1−~∇2φ1−2~∇·(Ωvxyˆ−Ωvyxˆ)ei(kz−ωt) = −ρ0 1 ρ0~∇2P1−~∇2φ1 ~2 ρ0~2φ1

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since we know that

P1=v2sρ1 ∇2φ1=4πGρ1

we can just plug this in to find ∂2ρ1

∂t2 =−v 2

s~∇2ρ1+ρ04πGρ1

since we also know that

ρ1=ρ1ei(kz−ωt) we find ∂2ρ1 ∂t2 = −ω 2ρ1ei(kz−ωt)=−ω2ρ1 ~2ρ1 = (−k)2ρ1 ei(kz−ωt) =k2ρ1 therefore we find −ω2ρ1=−v2 sk2ρ1+ρ04πGρ1

thus we find the dispersion relationship to be ω2=

v2sk2−ρ04πGρ1

(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 ∂2ρ1 ∂t2 =−ρ0 ∂(~∇·~v1) ∂t =−ρ0~∇·( 1 ρ0~∇P1−~∇φ1−2~Ω×~v1) we need to solve for

~ Ω×~v1 = ~v1·∇×~Ω−~Ω·~∇×~v1 = −~Ω·~∇×~v1 so we find ∂2ρ1 ∂t2 =−ρ0 ∂(~∇·~v1) ∂t =−ρ0~∇·( 1 ρ0~∇P1−~∇φ1−2(−~Ω·~∇×~v1)) but we know that

∂~v1 ∂t =

1

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so ~×∂~v1 ∂t = −2~∇×~Ω×~v1=−2~∇×(Ωvxyˆ−Ωvyxˆ) =−2Ω −∂vx ∂z xˆ− ∂vy ∂z yˆ+ ∂ vx ∂x − ∂vy ∂y ˆz = −2Ω∂v∂xxˆz= d dt∇×~v1 we also know that

∂ρ1 ∂t = −ρ0~∇·~v1=−ρ0 ∂v x ∂x + ∂vy ∂y + ∂vz ∂z = ρ0 2Ω ∂vx ∂x and so ρ1= ρ0 2Ω(~∇×~v1)z therefore ~·(~×~v1) =(×~v1)z= 2Ω2ρ1 ρ0 thus ∂2ρ1 ∂t2 =−ρ0 1 ρ0∇2P1−∇2φ1−2 2Ω2ρ1 ρ0 and as before −ω2ρ1 = −∇2P 1+ρ0∇2φ1+2ρ0 2Ω2ρ1 ρ0 −ω2 = −v2 sk2+ρ04πG+4Ω2

Thus we find the dispersion relationship to be ω2=v2

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The Formation of Galaxy Structure and Evolution of

Morphologies

Rastika’s

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 Lz , 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 (F606) 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)

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4. Corralating galaxy morphology to Sersic index.

References

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