ASTR 330 Problem Set 04: Extrapolating the Mass-Luminosity-Lifetime Scaling to a Hypothetical Massive Star

OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 19.3.

Problems

1. Extrapolated luminosity 20 points

Using L ≈ M^3.5, predict the luminosity of a hypothetical 10 M⊙ main-sequence star (an extrapolation exercise, not a specific measured real star).

2. Extrapolated lifetime 15 points

Using t_MS = 10 Gyr × M/L, compute this hypothetical star's main-sequence lifetime from your Problem 1 luminosity, and express the result in millions of years (Myr).

3. Comparison to Sirius A 15 points

Compare your Problem 2 lifetime to Sirius A's computed lifetime (0.812 Gyr, Lab 04), and explain, using the M^-2.5 scaling, why a star only about 4.8 times more massive than Sirius A has a dramatically shorter lifetime.

4. Limits of the extrapolation 20 points

The n=3.5 mass-luminosity exponent is only an approximation, and it is known to be less accurate at high stellar masses (where radiation pressure becomes important, per Lecture 02/PS 01). Explain in one paragraph why this problem set's answer should be treated as an order-of-magnitude estimate rather than a precise prediction, and identify one specific physical effect (from this course) that would need to be included in a more accurate treatment.

Due and Scope

Submit a complete derivation with equations, units, labeled quantities, and a short narrative interpretation for each problem. Show every step; a correct final number without a visible derivation receives partial credit at most, and quoting a lecture number without adapting it to this problem's specific inputs receives no credit for that step.

OpenStax Companion Reading

OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 19.3.

References and Data Sources