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CMB temperature at different ages
A discussion elsewhere lead me to wonder what the temperature
of the CMB was at about the time that Jupiter was forming. (I was wondering how the different background temperature would affect the rate at which the initial gravitational infall energy was radiated.) I'd guess that the temperature is declining exponentially, so I should be able to work out an expression from the decomposition temperature of hydrogen (ca. 5,000 K ? ), the present age of the universe (13.7 Gyr) and the present CMB temperature (3.7 K). But it would only be a guess. Does anyone have an expression that would give the CMB's temperature as a function of time - either time after the hydrogen recombination/ big bang, or back from the present day? Doesn't have to be too accurate - give or take a factor of two would be fine. -- Aidan Karley, Aberdeen, Scotland, Location: +57d10' , -02d09' (sub-tropical Aberdeen), 0.021233 Written at Mon, 21 Nov 2005 18:05 GMT |
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CMB temperature at different ages
Aidan Karley .group wrote:
A discussion elsewhere lead me to wonder what the temperature of the CMB was at about the time that Jupiter was forming. (I was wondering how the different background temperature would affect the rate at which the initial gravitational infall energy was radiated.) I'd guess that the temperature is declining exponentially, so I should be able to work out an expression from the decomposition temperature of hydrogen (ca. 5,000 K ? ), the present age of the universe (13.7 Gyr) and the present CMB temperature (3.7 K). But it would only be a guess. Does anyone have an expression that would give the CMB's temperature as a function of time - either time after the hydrogen recombination/ big bang, or back from the present day? Doesn't have to be too accurate - give or take a factor of two would be fine. At the epoch from which we now see radiation at redshfift z, the CMB had temperature 2.73(1+z). The time-z mapping depends on the cosmological parameters, and doesn't necessarily have a nice closed form. Ned Wright's calculator at http://www.astro.ucla.edu/~wright/CosmoCalc.html will deliver lookback times (which is what you need) for various z and cosmologies. For the popular WMAP "consensus" values, 4.6 Gyr back corresponds to z=0.44, for a CMB temperature of 3.9 K. Bill Keel |
#3
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CMB temperature at different ages
In article , Aidan Karley
.group writes: A discussion elsewhere lead me to wonder what the temperature of the CMB was at about the time that Jupiter was forming. (I was wondering how the different background temperature would affect the rate at which the initial gravitational infall energy was radiated.) I'd guess that the temperature is declining exponentially, Why? Exponential decline of temperature is associated with flow of heat from a warmer to a cooler body (IIRC this is known as "Newton's law of cooling"), which is not the case here. so I should be able to work out an expression from the decomposition temperature of hydrogen (ca. 5,000 K ? ), the present age of the universe (13.7 Gyr) and the present CMB temperature (3.7 K). But it would only be a guess. What is the relation between the decomposition temperature of hydrogen and the formation of Jupiter? Jupiter didn't form as soon as hydrogen could exist. Does anyone have an expression that would give the CMB's temperature as a function of time - either time after the hydrogen recombination/ big bang, or back from the present day? Doesn't have to be too accurate - give or take a factor of two would be fine. The temperature is inversely proportional to the scale factor. Thus, if the universe doubles in size, the temperature is half of what it was. If you know the epoch at which Jupiter was being formed, you can then work out the increase in the scale factor since then, and from that can easily calculate the temperature. The difficult part is the increase of scale factor with time. This depends on the cosmological parameters. However, they are relatively well known. Ned Wright's cosmology calculator might be interesting to look at. For some theory on how the temperature of the CMB evolves, see the first part of chapter 8 of Berry's COSMOLOGY AND GRAVITATION. Interestingly, since spectra are influenced by the ambient temperature, one can actually OBSERVE what the temperature was at high redshift, rather than just calculating it. The present temperature is approximately 2.7, not 3.7. Very roughly, Jupiter formed when the universe was about 2/3 of its present age. Since then, the rate of expansion has, on average, been roughly linear (to a first approximation). So, roughly, 4 K. |
#4
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CMB temperature at different ages
In article , Phillip
Helbig---remove CLOTHES to reply wrote: I'd guess that the temperature is declining exponentially, Why? It was a guess. I Am Not A Physicist. What is the relation between the decomposition temperature of hydrogen and the formation of Jupiter? Jupiter didn't form as soon as hydrogen could exist. It was to give one fixed point for getting the equation of the (guessed) exponential. Present-day being the other fixed point. Age of the planets/ Solar system/ Sol being the variable. Out pops the CMB temperature as (present temperature)*(1+z), moving the problem to getting a "z" value. (Thanks to others, William & Steve, for that.) The temperature is inversely proportional to the scale factor. Thus, if the universe doubles in size, That's what was making me aware that I was guessing with an exponential - doubles in linear dimension, in volume, in 4-d hypervolume? IANAP, as above. The difficult part is the increase of scale factor with time. As Mrs Beaton put it a century or so ago, "first catch your rabbit". Ned Wright's cosmology calculator might be interesting to look at. Reading the FAQ in another window just now. Anyway, CMB temperature around 3.9 K when Jupiter was formed, so to answer my original speculation, there's not going to be a lot of difference in the cooling curve of Jupiter when compared to a simple model of a blob of matter sitting in a constant-temperature universe. Which is what I'd expected. But it's nice to be able to put numbers to it. -- Aidan Karley, Aberdeen, Scotland, Location: +57d10' , -02d09' (sub-tropical Aberdeen), 0.021233 Written at Tue, 22 Nov 2005 03:55 GMT |
#5
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CMB temperature at different ages
In article , Aidan Karley
.group writes: The temperature is inversely proportional to the scale factor. Thus, if the universe doubles in size, That's what was making me aware that I was guessing with an exponential - doubles in linear dimension, in volume, in 4-d hypervolume? IANAP, as above. Here, linear dimension. |
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