Multi-Wavelength Observations of Cluster

[rstein66]

Homework Statement

Hi, I am having difficulties with a first-year astronomy question that was issued on a final a few years back.
A group of astronomers make various observations of a cluster spanning 10arcmin (600arcsec) on the sky. X-ray astronomers found the cluster to be full of hot gas, with λmax=0.0377nm.
Optical astronomers found the bolometric apparent magnitude of the whole cluster to be, m=10.4.
The mass to light ration of the stellar populations in the galaxies in the cluster was equal to 2. Meanwhile, radio astronomers found the neutral hydrogen line associated with the cluster (λHI)=27.3cm. Finally, for this problem use the Hubble constant of 70km/s/Mpc and the density of the intercluster medium to be equal to 10^-27g/cm^3

Find the following:
1. The redshift, z and distance to the cluster d, in pc.
2. The radius of the cluster r, in pc.
3. The mass of the hot gas in the cluster, Mgas.
4. Luminosity and mass of the stellar content, in solar units.
5. Cluster's total mass Mtot
6. Percentage of cluster's mass in dark matter.

Homework Equations

Stated in the question it says use basic formula and do not use modified inverse square laws for distant galaxies for brightness or size.
This question applies many formulas, I am not sure which to use for every part.

The Attempt at a Solution

1. z(of hydrogen line, using normal atomic hydrogen radius of 21cm found in textbook):
z=(λs-λr)/λr = (27.3cm-21.0cm)/21.0cm = 0.3

d=1/p = 1/600" = 0.00167pc.

2. Not sure what to do here.

3. Mgas= (r(Vgas)^2)/G
Vgas(X-ray)=140(sqrt(T)), where T is found by 2.9e6/λmax to be 7.69e7K. Thus, Vgas is equal to 1.23e6m/s, and therefore Mgas = r(1.23e6)^2/6.67e-11

4. L/Lsun=100^(4.75-M)/5
M=m+5-5log(d)
M=10.4+5-5log(0.00167) = 29.3, thus L=1.23e-5Lsun?
I don't know if this was done correctly?

5. Not sure what to do here.
6. I believe that Mtot-Mstars=Mdark matter then I just find its percentage but still don't have either Mtotal or Mstars.

Thanks so much!

 

[lightgrav]

1. the reason to find z first is so you can get the distance from that red-shift (with Hubble)
2. now that you know the distance (from redshift), and the angular size, find the real size.

 

[rstein66]

ah so for 1.
z=0.3
v=Ho*d, thus d=v/Ho, where v=cz

so 3e5km/s(0.3)/70km/s/MPC = 1286Mpc

Then for 2, using the formula: 2.5e5"(r/d)=theta(in arcsec)

r=1286e6pc(600") / 2.5e5" = 3.09Mpc

Did I do this correctly, I edited because I used a wrong formula

Continuing with this data i plugged in the radius for the mass of the gas formula and found it to be 2.2e45kg which seems to high.

And I think I know how to get the total cluster mass using its density stated and the formula density, n= m/v which I converted the radius in Mpc to m and used the volume of a sphere and made sure to change the n to kg/m^3 where the Mtot ended up being 3.62e41kgs which is somehow less then Mgas?

 

[rstein66]

Ok, disregard my previous post, I think I got it. If anyone could verify my answers and such I would really appreciate it.

1. z=(λshift,hydrogen - λrest,hydrogen)/λrest, hydrogen
z=27.3cm-21cm/21cm , z= 0.3

v=Ho*d, d=v/Ho where v is the recession velocity which is equal to c*z.

v=cz=3e5km/s(0.3)=90000km/s, thus d=(90000km/s)/70km/s/Mpc = 1286Mpc

2. 2.5e5"(R/d)=θ(in arcsec")
600"/2.5e5" = R/1286Mpc ... R=3.09Mpc (Stated in question it says this is "the gigantic Pandora cluster")

3. Mgas=R(Vgas)^2 / G , where Vgas is found with the X-ray data, using the formula Vgas=140(sqrt(T)) where T=2.9e6nm/0.0377nm = 7.69e7K
Thus Vgas = 123e4m/s and Mgas is equal to (9.53e22m<---[3.09e6pc])(1.52e12m/s)/6.67e-11
...Mgas=2.16e45kg=1.08e15Msun

4. L/Lsun=100^(4.75-M)/5
M=m+5-5log(d) = 10.4+5-5log(1286e6pc)
M=-30
...L=9.088e15Lsun

Thus, L/Lsun=(M/Msun)^3.5, L^-3.5=Mass of stars = 9731Msun

5.Mgalaxy,total=rv^2/G , have radius from before and use v as the recession velocity
...Mgalaxy,total=5.787e16Msun

6. (Mgalaxy,total)-Mstars-Mgas=Mdark matter, which ends up being ~98% of cluster's mass which according to my lecture notes seems accurate of clusters.

I am new to this site so I am not too sure how to write the formulas neatly and such but if you want me to expand on anything really just ask, I kinda just summed up my work.

 

[HalfLostAndSearching]

Hi there,

I would first like to commend you for attempting to solve this challenging astronomy problem. It shows your dedication and problem-solving skills. I will try to provide a response to the best of my abilities and guide you through the steps to solve this problem.

1. To find the redshift and distance to the cluster, we can use the formula z=v/c, where v is the recessional velocity of the cluster and c is the speed of light. The recessional velocity can be calculated using the Hubble's law, v=H0*d, where H0 is the Hubble constant and d is the distance to the cluster. Plugging in the given values, we get v=70*0.00167=0.1169 km/s. Using this value of v, we can find the redshift, z=0.1169/3e5=3.9e-4. To find the distance, we can use the formula d=c*z/H0, where c is the speed of light and H0 is the Hubble constant. Plugging in the values, we get d=3e5*3.9e-4/70=1.7 Mpc=1.7*10^6 pc.

2. The radius of the cluster can be found using the angular size of the cluster (10 arcmin) and the distance to the cluster (1.7 Mpc). We know that 1 arcmin=1/60 degrees and 1 degree=pi/180 radians. Therefore, the angular size of the cluster in radians is 10/60*pi/180=pi/1080. Using the formula r=d*tan(theta), where d is the distance to the cluster and theta is the angular size in radians, we get r=1.7*10^6*tan(pi/1080)=1.7*10^6*0.00005236=89.1 pc.

3. To find the mass of the hot gas in the cluster, we can use the formula M=v^2*r/G, where v is the velocity of the gas, r is the radius of the cluster, and G is the gravitational constant. We already calculated the velocity of the gas in the attempt at a solution, but we need to convert it to km/s, which gives us 1.23 km/s. Plugging in the values, we get M=(1