Compare fuel inventory with average power

Separate inventory from power

A reservoir can contain a great deal of usable energy and still be exhausted quickly if the prescribed output is large. This experiment asks how long a stipulated hydrogen inventory could support a stipulated average photon power. It does not calculate a star’s actual lifetime, its remaining age or its present evolutionary stage.

Write two headings on paper: available energy in joules and demanded power in joules per second. Dividing the first by the second gives seconds. Before calculating, predict what happens when only the inventory doubles, then when only the power doubles. These are different comparisons even if both are described loosely as making the star larger or brighter.

Calculate the hydrogen actually available

The hypothetical total stellar mass M is fixed at 2.0 × 10³⁰ kg. It is a chosen round value, not an exact solar mass. The available-region control supplies a percentage a; its fraction is q = a/100. That region initially has hydrogen mass fraction X = 0.70. Hydrogen fuel mass is therefore Mf = XqM.

At the default a = 10, q = 0.10. The available region contains 2.0 × 10²⁹ kg of material, of which 1.40 × 10²⁹ kg is hydrogen fuel. The remaining material is not an additional hydrogen supply. All of the specified hydrogen is assumed convertible through the restricted net chain, with no calculation of mixing or spatial fuel gradients.

Make a three-row ledger for total mass, available-region mass and hydrogen mass. Compare 5, 10 and 20 percent at fixed photon demand. Predict the intermediate quantities before using the lab. A core-region fraction and a hydrogen fraction inside that region are two separate factors, not competing descriptions of the same percentage.

Turn fuel into a conditional duration

From the atomic mass and energy ledger, the deposited energy per kilogram of hydrogen is approximately 6.271 × 10¹⁴ J/kg. Multiplying by the default fuel inventory gives approximately 8.779 × 10⁴³ J. Deposited energy excludes the fixed mean energy escaping in pp neutrinos.

The selected average photon power is a multiple of nominal solar luminosity L₀ = 3.828 × 10²⁶ W, defined in the IAU nominal-constant resolution. For one L₀, duration τ = Edeposit/Lγ is approximately 7.268 billion model years. One displayed year is 31,557,600 seconds. A billion years is one Gyr; it is a unit conversion, not an age determination.

This ratio assumes deposited nuclear energy eventually supports the stated average photon luminosity, without net stored-energy changes, mechanical losses or another energy source. The underlying pp-chain context comes from Lawrence Berkeley National Laboratory. Energy per completed chain does not supply the missing stellar-structure and reaction-rate physics.

Make paired comparisons

At 20 percent and one L₀, available energy and duration double to approximately 14.535 Gyr; the required reaction rate stays fixed. At 10 percent and two L₀, duration halves to approximately 3.634 Gyr while the required rate doubles. Compare each with the reference, which always uses the same selected reservoir and one L₀.

Capture a third case: 20 percent and two L₀. Doubling both reservoir and demand restores the default duration. Equal duration therefore does not imply equal fuel inventory or equal luminosity. Explain the cancellation using a fraction on paper before consulting the calculated result.

The sensitivity chart compares separate budgets across available-region percentages. It is not a time history of a core expanding. Its fixed scale includes the long mathematical duration at 20 percent and 0.1 L₀, approximately 145.35 Gyr. That output is a consequence of chosen inventory and power, not a prediction that a corresponding real star survives that long.

Test the zero-fuel boundary

Set the available region to zero. Every positive luminosity option now has zero possible completions, energy and support duration. The proposed power and its required reaction rate are not achieved output. Do not interpret a positive demand label as evidence that a fuel-free system is shining through this mechanism.

Transfer: diagnose a lifetime claim

A report finds the same budget duration for two selected cases and concludes that both stars have the same remaining main-sequence lifetime. Give the valid numerical conclusion and identify two missing physical ingredients before opening the worked response.

Check the fusion-budget transfer

The ratios of assumed deposited inventory to prescribed average photon power are equal. Actual remaining lifetime would require evidence about accessible remaining hydrogen, mixing and evolving luminosity, together with a stellar-structure/evolution model. Other reactions and energy storage may matter. The public calculation establishes a conditional fuel budget, not age or independent mastery.