University Physics IV · topic UPIV-13.09
Fusion, the Gamow peak, and stellar burning
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Fusion and the Gamow peak: predict a finite change
The Gamow peak is the band of collision energies — high on the Maxwell tail, yet far below the Coulomb barrier — that contributes almost all of a thermonuclear reaction's rate. Use the supplied topic model Y_out = 3*(b*b*(0.08617*t)^2/4)^(1/3)/(0.08617*t), where Y_out is EXPONENT τ = 3E₀/kT (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. t is Plasma temperature T, measured in 10⁶ K. Keep b (Gamow constant b) = 22.2 √keV fixed. For this exercise use only 8.71875 10⁶ K ≤ t ≤ 36.2813 10⁶ K on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at t = 14.7825 10⁶ K and 18.6412 10⁶ K. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δt across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
EXPONENT τ = 3E₀/kT against t
Predictions from the supplied Fusion and the Gamow peak model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): EXPONENT τ = 3E₀/kT
Horizontal axis (x): t: Plasma temperature T / 10⁶ K
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| t: Plasma temperature T / 10⁶ K | EXPONENT τ = 3E₀/kT |
|---|---|
| 8.71875 | 16.42100756910092 |
| 15.609375 | 13.52350828777459 |
| 22.5 | 11.97173339925724 |
| 29.390625 | 10.95170160759452 |
| 36.28125 | 10.209170630951395 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of t separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At t = 14.7825 10⁶ K, substitute into Y_out = 3*(b*b*(0.08617*t)^2/4)^(1/3)/(0.08617*t): Y₀ = 13.7711.
- At t = 18.6412 10⁶ K, the same fixed-parameter model gives Y₁ = 12.7466.
- Signed change: ΔY_out = Y₁ − Y₀ = -1.02454; the input change is Δt = 3.85875 10⁶ K.
- Average sensitivity is ΔY_out/Δt = -0.265511 1 per 10⁶ K. Its negative sign means that this response decreases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 13.7711, Y₁ = 12.7466; ΔY_out = -1.02454; average sensitivity = -0.265511 1 per 10⁶ K.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Fusion and the Gamow peak: infer an input with uncertainty
The Gamow peak is the band of collision energies — high on the Maxwell tail, yet far below the Coulomb barrier — that contributes almost all of a thermonuclear reaction's rate. Use the supplied topic model Y_out = 3*(b*b*(0.08617*t)^2/4)^(1/3)/(0.08617*t), where Y_out is EXPONENT τ = 3E₀/kT (dimensionless). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793. t is Plasma temperature T, measured in 10⁶ K. Keep b (Gamow constant b) = 22.2 √keV fixed. For this exercise use only 8.71875 10⁶ K ≤ t ≤ 36.2813 10⁶ K on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 11.477756, with a bounded reading uncertainty ±0.19029. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown t by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00275625 10⁶ K, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
EXPONENT τ = 3E₀/kT against t
Predictions from the supplied Fusion and the Gamow peak model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): EXPONENT τ = 3E₀/kT
Horizontal axis (x): t: Plasma temperature T / 10⁶ K
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| t: Plasma temperature T / 10⁶ K | EXPONENT τ = 3E₀/kT |
|---|---|
| 8.71875 | 16.42100756910092 |
| 15.609375 | 13.52350828777459 |
| 22.5 | 11.97173339925724 |
| 29.390625 | 10.95170160759452 |
| 36.28125 | 10.209170630951395 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(t) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(8.71875) = 16.421 and Y_out(36.2813) = 10.2092. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives t_line = 8.71875 + (11.4778 − 16.421)(36.2813 − 8.71875)/(10.2092 − 16.421) = 30.6524 10⁶ K. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives t = 25.5319 10⁶ K; substituting back gives Y_out = 11.477756.
- The interpolation error in the input is |t_line − t| = 5.12055 10⁶ K. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 11.287466 and 11.668046. Sort the roots even if the response decreases: 24.303 10⁶ K ≤ t ≤ 26.8451 10⁶ K.
- The lower and upper input deviations are 1.22891 10⁶ K and 1.31318 10⁶ K. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer t ≈ 25.5319 10⁶ K; compatible interval [24.303, 26.8451] 10⁶ K. Endpoint interpolation gives 30.6524 10⁶ K.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.