Dark Matter and Galaxy Rotation

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Questions and Answers

Why was dark matter first theorized?

Because the observed mass of galaxies was much higher than predicted by the Cosmic Microwave Background.
Because star velocities at the periphery of galaxies were higher than expected based on visible mass. (correct)
Because scientists wanted to find out what galaxies are made of.
Because the brightness of galaxies was much lower than expected.

What is the fundamental discrepancy at the heart of the galaxy rotation problem?

The observed velocities of stars contradict Newton's second law of motion.
The calculated velocity of stars based on luminosity is much higher than their actually measured velocity.
The distribution of luminosity in a galaxy is not symmetric, violating a key assumption in calculations.
The mass of a galaxy estimated from its luminosity differs significantly from the mass estimated from the velocities of its stars. (correct)

In the derivation of the relationship between mass and velocity, what assumption is made about the location of a galaxy's mass?

The galaxy's mass is concentrated in a flat disk.
The galaxy's mass is concentrated in its spiral arms.
The galaxy's mass is approximated as being located at its center. (correct)
The galaxy's mass is uniformly distributed throughout its volume.

In the derivation of the relationship between mass and velocity, the acceleration contribution is derived from the assumption of what type of orbit?

Circular motion. (B) Signup and view all the answers

During the mathematical derivation relating a star's velocity to the galaxy's mass, which of the following variables are simplified out?

The mass of the star and the radius of its orbit. (B) Signup and view all the answers

How do galaxy rotation curves provide evidence for dark matter?

They show that the observed velocity of stars remains constant or increases with distance from the galactic center, which cannot be explained by visible matter alone. (C) Signup and view all the answers

What effect does adding a dark matter term to the mass of the galaxy have?

It results in an increase in the expected velocity of stars at the periphery. (D) Signup and view all the answers

Beside galaxy rotation curves, what other evidence supports the existence of dark matter?

Studies of galaxy clusters and the Cosmic Microwave Background. (C) Signup and view all the answers

How can a galaxy's mass be inferred?

By measuring the total brightness (luminosity) of the galaxy, or measuring the velocities of stars. (A) Signup and view all the answers

Which of the following describes the relationship between F, m, v, and r in the context of gravitational force?

$F = mv^2/r$, where F is force, m is mass, v is velocity, and r is radius. (B) Signup and view all the answers

Flashcards

Galaxy Rotation Problem

A discrepancy where mass inferred from a galaxy's luminosity differs from that calculated using star velocities.

Dark Matter

Theorized substance making up the 'missing mass' in galaxies, inferred from gravitational effects.

Law of Gravity

Attractive force between two objects, proportional to their masses and inversely proportional to the square of the distance.

Newton's Second Law

Force equals mass times acceleration (F=ma).

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Rotation Curve

Plot of orbital velocities of stars or gas in a galaxy versus their radial distance from the galaxy's center.

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Center of Mass

The principle that the total mass of an object can be considered as concentrated in one point.

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Gravitational Constant (G)

Constant that quantifies the strength of the gravitational force.

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Study Notes

Dark Matter and the Galaxy Rotation Problem

Dark matter was first theorized in the 1930s due to the galaxy rotation problem.
The galaxy rotation problem arises from discrepancies in measuring a galaxy's mass using different methods.
One method infers mass from the total brightness (luminosity) of the galaxy.
Another method measures the velocities of stars, especially those at the galaxy's periphery.
These two methods should yield similar results but do not.
The expected and observed plots of mass distribution should overlap, but they don't, indicating missing mass.
The Milky Way is symmetric and spiral bar-shaped.

Deriving the Expected Relationship Between Mass and Velocity

The relationship between the mass of a galaxy and the velocity of stars within it can be derived using high school physics principles.
Newton's second law (F=ma) and the law of gravity are used.
Newton's second law expresses the relationship between force, mass, and acceleration.
The law of gravity describes the attraction between two bodies, in this case, a star and the galaxy.
The galaxy's mass can be approximated as being located at the center due to the principle of center of mass.
The orbit of a star around the galaxy is assumed to be circular due to the galaxy's symmetry.
The acceleration used in Newton's second law is derived from circular motion.

Mathematical Derivation

The force expressed by Newton's second law and the law of gravity are equated.
The acceleration for circular motion is plugged into Newton's second law.
The derivation involves equating the expressions from Newton's second law and the law of gravity.
The mass of the star and one of the radii are simplified out.
This yields a relationship between the mass of the galaxy inside the star's orbit and the star's velocity.
For a star further away, the mass between the two orbits is negligible compared to the galaxy's total mass.
The mass of the galaxy is treated as a constant.
G, the gravitational constant, is also a constant.
A relationship is extrapolated relating the velocity of stars at the periphery to their distance from the galaxy's center, assuming constant mass.

Understanding the Galaxy Rotation Curves

The galaxy M33's rotation curve illustrates the discrepancy.
The graph plots velocity (y-axis) against radius (x-axis).
One plot is derived from total luminosity (expected) and the other is measured from star velocities (observed).
The observed velocities are significantly higher than expected.
This means that with a greater velocity there is a greater gravitational Force (F = mv^2/r) as such there is greater mass
The discrepancy suggests the existence of non-observable (dark) matter.
Adding a dark matter term to the mass in the equations accounts for this discrepancy.
Extra term added to the mass of the galaxy results in an increase in velocity.

Evidence for Dark Matter

The galaxy rotation problem provides justification for dark matter's existence.
Other experiments, like the Cosmic Microwave Background and studies of galaxy clusters, offer further evidence.