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Galaxy expansion & cosmological constant 
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#1
Feb1212, 02:27 PM

P: 442

Hi Astro forum... I have got something to clarify which came across my mind as I was trying to answer few problems.So without any further delay , let's start :
For simplicity we will assume that dark matter doesn't influence the center of mass of galaxy A. Now let's say galaxy A has mass 'M' concentrated at the center with a separation distance of 'r' from the edge. In this case with simple arithmetic manipulation we get : [itex]V_{escape}[/itex] = [itex]\sqrt{GM/r}[/itex] Now I know that at shorter separation distances gravitational forces overwhelm ( i.e are greater) than cosmological constant , and then using Hubble's law v = [itex]H_{0}[/itex] r( assuming redshift is inteperted as doppler shift) .. so far it all makes sense... But if I am asked to prove that Vesc >> v ... I was thinking of something as following: [itex]V_{escape}[/itex] = [itex]\sqrt{GM/r}[/itex] Squaring both sides to get: [itex]V_{escape}^2[/itex] = [tex]2GM/r[/tex] which gives r = [itex]2GM/ V_{escape}^2[/itex] Now as said previously v = [itex]H_{0} r[/itex] so v = [itex]H_{0} 2 GM / V_{escp^2} [/itex] [itex]V_{escp^2} = H_{0} *2 GM / v [/itex] Gives us [itex]V_{escp} \propto[/itex] [itex]v^{1/2}[/itex] Is this suffice enough to prove Vesc >> v ? am I on the dot on this one ? Any input is appreciated. ibysaiyan EDIT: Oh why are my latex command showing up.. hm.. edit2: latex error fixed 


#2
Feb1212, 03:58 PM

Sci Advisor
P: 2,193

I have no idea what it is you're trying to prove here, but whatever it is, you're going about it wrong. First off, you state that you want to prove [itex] V_{esc} \gg v[/itex], what is [itex]v[/itex]? The characteristic velocity of stars in the galaxy? I have absolutely no idea what it is you're trying to derive, it just looks like mindless symbol manipulation and combination of equations which leads me to the second point...
Doing anything on a galactic scale, you cannot use Hubble's velocity relation. It's not merely that the recessional velocities predicted by it will be much smaller than typical velocities of galactic motion (is this what you're trying to prove???), hubble expansion DOESN'T HAPPEN in gravitationally bound clusters. Furthermore, this is completely separate from the cosmological constant! In a traditional LambdaCDM model, the density of dark energy is constant, and gravitationally bound objects (like our galaxy) will stay gravitationally bound indefinitely. (I say traditional because there are some models in which Lambda is somehow a function of time, and grows, leading to 'big rip' scenarios). As a final point, your equation for escape velocity is incorrect: [tex]v_{esc}=\sqrt{\frac{2GM}{r}}[/tex] While [tex]v_{orbit}=\sqrt{\frac{GM}{r}}[/tex] Where this [itex]v_{orbit}[/itex] is the velocity of a circular orbit of radius r. 


#3
Feb1212, 04:46 PM

P: 442

I think it would be helpful if I post the actual question than wrongly restating it. When you say: 1) Does that refer to a cluster of galaxies ? 2) if dark energy density stays constant does that mean it's homogeneous , as in not being concentrated at a point ? Here's the question: http://i42.tinypic.com/8wzh20.png P.S : 'v' is the recessional speed of the galaxy. 


#4
Feb1212, 07:31 PM

Sci Advisor
P: 2,193

Galaxy expansion & cosmological constant
Now that I understand the question though, your calculation in post 1 is incorrect. There is a scaling relationship between the two equations: on some scales, one velocity dominates, while on others, the opposite is true. Therefore, the way to do it is to calculate the characteristic velocities of both, and then compare them. (i.e. assume typical scales for M and r). 


#5
Feb1212, 09:08 PM

P: 442

Do I find a set of equations relating recessional speed using Hubble's constant and v_esc in the above equation ? Apart from the escape velocity given ~ 103 c (300km/s) and Hubble's constant ~ 75km/s/Mpc there isn't much given. Hm.. EDIT: I took some arbitrary points and the results which i ended up with are: Hubble speed of 757*10^3 km/s for separation distance of 2.6*1^17 km... when I plug the same distance in v_escapre equation.. it gives 300km/s ... 


#6
Feb1212, 09:18 PM

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#7
Feb1212, 10:11 PM

P: 442




#8
Feb1312, 01:30 PM

P: 442

So does anyone think my values are sensible ?



#9
Feb1312, 02:25 PM

Sci Advisor
P: 2,193




#10
Feb1312, 02:36 PM

P: 442




#11
Feb2212, 07:10 PM

P: 16

Galaxy expansion??



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