Help Needed: Calculating Universe Temperature at Star System Scale

[smitty8371]

Hi I am new to this site and I really need help I missed this class and of course my book doesn't have this formula so if anyone could tell me it that would be great.

The temperature of the universe is now 2.725 K, and its scale size is 100 Mly.
How hot was the universe when its length scale was the size of a star system (12billion km)?

 

[e.bar.goum]

That's a fun one.

I think the way I did this question was to recall that the temperature scaled linearly with redshift.

 

[smitty8371]

are you saying use 1+z=a1/a2? sorry I am still really confused
this is not a homework problem its just something i need to know for my test

 

[e.bar.goum]

Not what I was thinking of, but that might also work!

I was thinking of the fact that black body radiation follows Planks law, ie T*(z)=T_0*(1+z) But you might have to prove that if you haven't been taught it in class.

 

[smitty8371]

the only issue I am having with this is i only know the past scale size and the current scale size and temp but those other equations need wavelength

 

[e.bar.goum]

Wavelength? You shouldn't.

Use the scale factors to find the redshift, then you can get the temperature.

 

[smitty8371]

so then where do i plug in the scale factor. Do i use it for z

 

[e.bar.goum]

Like I said, use it to find the redshift.

 

[cepheid]

If a0 (the scale factor today) is 1, then in general a(t) = 1/(1+z(t)). In particular, for the two values you've been given a1 = (1/(1+z1)) and a2 = (1/(1+z2)). So you know the ratio of the (1+z)'s at the two times. As e.bar.goum has already pointed out, the radiation temperature scales inversely with a (or linearly with 1+z).

EDIT: I'm thinking of a dimensionless scale factor that represents the ratio of the separation of any two objects at time t to their separation now. You seem to be using some sort of dimensional scale length (i.e. one with units). The principle should be the same, but I don't know if the relationship between scale factor and redshift holds.

 

[e.bar.goum]

Sorry if I was being too obscure.

 

[smitty8371]

thanks i understand the concept of this its just where to put the numbers. I know that as the size decreases the temperature goes up that makes sense its the number crunching I am struggling with

 

[cepheid]

thanks i understand the concept of this its just where to put the numbers. I know that as the size decreases the temperature goes up that makes sense its the number crunching I am struggling with

Break it up into steps:

1. Let's just say that the scale factor a is the ratio of the universe's scale length at the time of interest to the scale length NOW. So what is the scale factor at the time when the scale length was only 12 billion km?

2. Given that scale factor, what is the redshift of light emitted at that time?

3. Given that redshift, by what factor does the temperature then differ from the temperature now.

 

[smitty8371]

would the scale factor be 1.268*10^-11
honestly could someone just show me the work to do it i know no one here wants to hear that but it would help me tremendously so then i can just remember the steps on how to solve it

 

[cepheid]

would the scale factor be 1.268*10^-11

Yes.

(That is the ratio of the two scale lengths.)

honestly could someone just show me the work to do it i know no one here wants to hear that but it would help me tremendously so then i can just remember the steps on how to solve it

No.

Now do step 2!

 

[smitty8371]

would the redshift be (1/1.268*10^-11)-1=z=about 7.88644*10^10
ok thank you i got the answer correct now then how would i go about finding the maximum wavelength

 

[e.bar.goum]

Maximum wavelength? I thought you were trying to find the temperature of the universe?

 

[smitty8371]

I was but the next part of my practice problem asks for a maximum wavelength for the previous problem

 

[smitty8371]

i was but its the next question on my practice problem sheet