In units G=1, circular speed is 4 at radius 3 in a spherical model. Find enclosed mass.
Answer and explanation
48
M=v²r/G=16×3.
Subjects
Cosmology
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Sample exercises
A flat rotation curve in the stated spherical model implies enclosed mass changes how with radius?
- It must fall as radius cubed
- Approximately proportionally to radius
- It must be zero
- It must be constant
Answer and explanation
Approximately proportionally to radius
With constant v, M(<r)=v²r/G grows linearly.
Which is an additional gravitational probe of mass distribution?
- Gravitational lensing
- Only a galaxy's name
- Only visible colour without a model
- A telescope's price
Answer and explanation
Gravitational lensing
Lensing responds to gravitational effects on light propagation.
Why is converting luminosity directly into total mass model-dependent?
Answer and explanation
Stellar populations have different mass-to-light ratios, and gas or nonluminous components can contribute. Luminosity alone does not uniquely fix mass.
Credit population models and unseen components.
At fixed circular speed in the spherical model, radius doubles. Find the inferred enclosed-mass factor.
Answer and explanation
2
M scales with r when v is fixed.
Explain why combining lensing and dynamics can be more informative than either alone.
Answer and explanation
The methods constrain gravitational effects through different data and assumptions. Agreement can strengthen or challenge a mass model while revealing systematics.
Credit complementary constraints, not a guarantee of unique truth.
Study notes
Galaxies, dynamics and dark-matter evidence
Galaxies contain stars, gas, dust and other gravitating components. Their morphology does not alone determine all their dynamical properties.
Advanced universityGalaxies, dynamics and dark-matter evidence · revision notes
Galaxies contain stars, gas, dust and other gravitating components. Their morphology does not alone determine all their dynamical properties.
Advanced universityRedshift, expansion and cosmological distances
Define z=(λobserved−λemitted)/λemitted. Redshift is dimensionless and can arise through different physical mechanisms.
Advanced universityRedshift, expansion and cosmological distances · revision notes
Define z=(λobserved−λemitted)/λemitted. Redshift is dimensionless and can arise through different physical mechanisms.
Advanced universityRedshift from a spectral line
This measured wavelength ratio alone does not select a velocity-distance model.
Advanced universityRedshift from a spectral line · formula reference
This measured wavelength ratio alone does not select a velocity-distance model.
Advanced universityScale factor from cosmological redshift
Apply the homogeneous expanding-universe cosmological-redshift model, not a local Doppler shift.
Advanced universityScale factor from cosmological redshift · formula reference
Apply the homogeneous expanding-universe cosmological-redshift model, not a local Doppler shift.
Advanced universityEnclosed mass from an idealized rotation speed
Assume circular motion about a spherically symmetric mass distribution; this is not valid for all geometries.
Advanced universityEnclosed mass from an idealized rotation speed · formula reference
Assume circular motion about a spherically symmetric mass distribution; this is not valid for all geometries.
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