Apparatus and Setup
This lab applies the mass-luminosity relation and main-sequence-lifetime scaling from Lecture 06 to a representative young open cluster whose brightest (most massive) main-sequence star has an inferred mass of 3.0 M☉, consistent with published turnoff masses for clusters of this type (e.g. the Pleiades-class age range); no specialized apparatus is needed, only the mass-luminosity relation itself.
Materials and Data
| Mass (M☉) | L = M^3.5 (L☉) | Main-sequence lifetime (Gyr) | Lifetime (Myr) |
|---|---|---|---|
| 0.5 | 0.09 | 56.5685 | 56568.5 |
| 1.0 | 1.00 | 10.0000 | 10000.0 |
| 3.0 | 46.77 | 0.6415 | 641.5 |
| 10.0 | 3162.28 | 0.0316 | 31.6 |
Procedure
- Verify the four rows in the Materials table by computing L = M^3.5 and t = 10 Gyr × M^−2.5 for each mass, showing your arithmetic for at least two of the four masses.
- A cluster's main-sequence turnoff mass is observed to be 3.0 M☉ (stars above this mass have already evolved off the main sequence; stars at or below it are still core-hydrogen-burning). Using the M^−2.5 scaling, compute the cluster's age.
- A second cluster has a turnoff mass of 0.9 M☉. Compute its age and compare it to the age of the universe (about 13.8 Gyr) — is this turnoff mass plausible for any real cluster? Explain using the concept of globular cluster ages from Lecture 06.
- Explain, quantitatively, why a 10 M☉ star's main-sequence lifetime is more than an order of magnitude shorter than a 3 M☉ star's, even though it has only about 3.3 times more mass (fuel).
- Using the first cluster's 3.0 M☉ turnoff mass and its derived age, estimate roughly how much longer the cluster's 0.5 M☉ stars (still far below the turnoff) will remain on the main sequence before they, too, evolve off it.
Analysis and Uncertainty
The exponent in L ∝ M^3.5 is itself an empirical fit with some scatter (real exponents range roughly 3-4 depending on mass range); state how a 10% uncertainty in the exponent would change your derived cluster age qualitatively (would the age estimate become more or less uncertain at higher turnoff mass, where the lifetime is most sensitive to the exponent?).
Deliverables
- Verified mass-luminosity-lifetime table with shown arithmetic for at least two masses.
- Cluster age calculation for the 3.0 M☉ turnoff case.
- Plausibility analysis for the 0.9 M☉ turnoff case.
- Remaining-lifetime estimate for the cluster's 0.5 M☉ stars.
Assessment Criteria
- Correct L and lifetime arithmetic for the verification rows.
- Correct cluster age for the 3.0 M☉ case (within rounding).
- Correct, well-reasoned plausibility judgment for the 0.9 M☉ case referencing the age of the universe.
- Correct comparative reasoning for the 10 M☉ vs. 3 M☉ lifetime scaling.
References and Provenance
- OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 22.1-22.3.
- Mass-luminosity relation and turnoff mass computed in
materials/ASTR230/src/generate_astr230_content.py.