Apparatus and Setup
Part 1 estimates the Sun's central pressure from the uniform-density hydrostatic-equilibrium scaling (Lecture 05) and compares it to the published standard-solar-model value. Part 2 applies the virial theorem's cluster-mass estimator (Lecture 06) to the globular cluster M15, using its published velocity dispersion and half-light radius (standard literature values, flagged for spot-check in ../reference-log.md).
Materials and Data
| Quantity | Value |
|---|---|
| Sun mass, radius | 1.989e+30 kg, 6.957e+08 m |
| Standard solar model central pressure (published) | 2.477e+16 Pa |
| Standard solar model core temperature (published) | 1.570e+07 K |
| M15 velocity dispersion σ (published) | 12.0 km/s |
| M15 half-light radius R_h (published) | 4.2 pc |
| M15 commonly cited dynamical mass (published) | 5.6e+05 M⊙ |
Procedure
- Using P_c ≈ (3/8π)GM²/R⁴, compute the Sun's estimated central pressure and compare it (percent difference) to the published standard-solar-model value.
- Using T ≈ μm_p GM/(5kR) with μ = 0.5 (fully ionized hydrogen), compute the Sun's estimated central/mean temperature and compare it to the published core temperature.
- Using M ≈ 5σ²R_h/G, compute M15's virial mass from its published velocity dispersion and half-light radius, and compare it (percent difference) to the commonly cited dynamical mass.
- State, for each of the three estimates above, whether the simple order-of-magnitude formula over- or under-predicts the more detailed/independently cited value, and propose one physical reason for the direction of each discrepancy (e.g., the Sun's real density profile is far more centrally concentrated than uniform; M15's true mass profile is not a single uniform sphere).
Analysis and Uncertainty
The uniform-density approximation used in both the hydrostatic-pressure and virial-temperature estimates ignores the fact that real self-gravitating systems (stars, clusters) are centrally concentrated, not uniform; state explicitly whether you would expect this to make the true central pressure/temperature higher or lower than the uniform-density estimate, and check whether your Part 1 results are consistent with that expectation.
Deliverables
- Sun central-pressure and central-temperature estimates with percent-difference comparisons to the published values.
- M15 virial-mass estimate with a percent-difference comparison to the published dynamical mass.
- Written discussion of the direction and physical cause of each discrepancy.
Assessment Criteria
- Correct unit conversions and arithmetic for all three order-of-magnitude estimates.
- Correct percent-difference calculations against the stated published comparison values.
- Physically sound (not merely stated) reasoning for the direction of each discrepancy, referencing central concentration.
References and Provenance
- OpenStax Astronomy 2e (local reference copy: references/openstax-astronomy-2e-extracted.txt), Chapter 15.4-15.5.
- M15 velocity dispersion, half-light radius, and dynamical mass: standard published globular-cluster literature values, not independently re-verified this session; flagged for spot-check against the Harris (1996, 2010 revision) Galactic globular cluster catalog in
materials/ASTR310/reference-log.md. - Standard solar model central pressure/temperature: standard published values, not independently re-verified this session.