Calculating Core Nuclear Reaction Rates for Stars Undergoing the CNO Cycle

[AStaunton]

Problem:

Two stars with identical core composition and density are undergoing nuclear fusion by the CNO cycle. Star A has a core temperature of 10% higher than star B. What is the ratio of their core nuclear reaction rates?

My solution:

as the energy generation rate q is proportional to (rho)T^16 I simply divided one by the other:

as density is the same for both stars, (rho) cancels... so left with:

(1.1T)^16/T^16=4.595

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My main query is: Did I use the correct equation, ie the equation I used is about the energy generation rate whereas the question talks about the nuclear reaction rate...I think however, that the energy generation rate is proportional to the reaction rate so whatever proportionality constant there is between them should just cancel out, but I am not entirely certain of this.

Any feedback appreciated.

 

[mark726]

Your solution is correct. The equation you used, q ∝ (ρT)^16, is known as the energy generation rate equation and it is used to calculate the rate at which energy is being generated in the core of a star through nuclear fusion reactions. The energy generation rate is directly proportional to the nuclear reaction rate, so using this equation is valid in this scenario.

The proportionality constant between the energy generation rate and the nuclear reaction rate is known as the energy generation efficiency, which is typically around 0.7 for the CNO cycle. However, as you correctly pointed out, this constant cancels out when comparing the ratio of the two stars' core nuclear reaction rates.

In summary, your solution is correct and you used the appropriate equation to solve the problem. Keep up the good work!