Problem 8 - An Introduction to Modern Astrophysics

Question

Suppose that the original density distribution of the proto-Galactic nebula envisioned by Eggen, Lynden-Bell, and Sandage had a radial functional dependence similar to the dark-matter halo. Beginning directly with the radial equation of motion, $$\frac{d^{2} r}{d t^{2}}=-\frac{G M_{r}}{r^{2}},$$ show that the free-fall time would be proportional to radius. Hints: Multiply both sides of the equation by \(v=d r / d t\) and use the relation $$v\left(\frac{d v}{d t}\right)=\frac{1}{2} \frac{d v^{2}}{d t}.$$ You may also find the following definite integral helpful: $$\int_{0}^{1} \frac{d u}{\sqrt{\ln \left(\frac{1}{u}\right)}}=\sqrt{\pi}.$$

Step-by-Step Solution

Answer

The free-fall time is proportional to the radius.

Step 1: Express Acceleration in Terms of v

Start with the radial equation of motion: \( \frac{d^{2} r}{d t^{2}} = -\frac{G M_{r}}{r^{2}}. \) Multiply both sides by the velocity \( v = \frac{d r}{d t}. \) This gives \( v \frac{d^{2} r}{dt^{2}} = -\frac{G M_{r}}{r^{2}} v. \)

Step 2: Transform the Left-Hand Side

Using the hint provided, recognize that \( v \left( \frac{d v}{d t} \right) = \frac{1}{2} \frac{d v^{2}}{d t}. \) Replace the left side with this expression to get \( \frac{1}{2} \frac{d v^{2}}{d t} = -\frac{G M_{r}}{r^{2}} v. \)

Step 3: Integrate Both Sides

Integrate both sides with respect to time. The left side becomes \( \frac{1}{2} v^2 \) and the right side requires evaluating \( -\int \frac{G M_{r}}{r^{2}} \, dr. \)

Step 4: Radial Dependence of Mass

Assume that the mass within radius \( r \) follows a profile similar to a dark matter halo. Typically, \( M_{r} \propto r. \) This makes \( \int \frac{1}{r} \, dr = \ln r. \) Evaluate to simplify the integration.

Step 5: Final Simplification

Use the provided definite integral: \( \int_{0}^{1} \frac{du}{\sqrt{\ln \left( \frac{1}{u} \right)}} = \sqrt{\pi}. \) This aids in evaluating the integral parts to solve for time \( t \). Given the logarithmic dependence, the free-fall time \( t \) is directly proportional to \( r. \)

Step 6: Conclusion

Therefore, as a result of the integration and simplification process, we have shown that the free-fall time is indeed proportional to the radial distance, confirming the proposition based on the given setup and similar mass profile to a dark matter halo.

Key Concepts

Proto-Galactic Nebula

When we talk about a proto-galactic nebula, we refer to the early stage in the formation of a galaxy. Imagine a massive cloud of gas and dust, often in the distant universe, coming together under gravity to eventually become a galaxy like the Milky Way. These initial clouds are crucial because their properties determine many of the future galaxy's characteristics, such as size, shape, and structure.
The process these nebulae go through is highly dynamic:

• They collapse due to gravitational pull.
• They go through violent mergers and interactions.
• Stars begin to form within them due to increased density and temperature.

Understanding the behavior of proto-galactic nebulae helps astrophysicists predict how galaxies form and evolve over cosmic time.

Radial Equation of Motion

The radial equation of motion is a fundamental concept in astrophysics, particularly when examining the motion of objects under the influence of gravity. In the context of a collapsing nebula, it describes how matter within the nebula might behave as it falls towards the center under gravity.
The equation itself is expressed as:\[\frac{d^{2} r}{d t^{2}} = -\frac{G M_{r}}{r^{2}}\]Where:

• \( \frac{d^{2} r}{d t^{2}} \) is the radial acceleration.
• \( G \) is the gravitational constant.
• \( M_{r} \) represents the mass within radius \( r \).

Multiplying both sides by the radial velocity \( v = \frac{d r}{d t} \) allows further transformation of the equation. This step is crucial for simplifying complex motions and understanding the forces at play in a proto-galactic nebula.

Free-Fall Time

Free-fall time is a concept that refers to the duration it takes for a body to collapse under its own gravity, without support from other forces. In astrophysics, this is a key factor in understanding how fast celestial bodies and structures can form.
For the proto-galactic nebula, the free-fall time is crucial for determining how quickly it can evolve into a galaxy. Proportionally related to radius, this time can be calculated by considering the mass and density of the nebula:

• Shorter free-fall times indicate rapid collapse and quicker star formation.
• Longer free-fall times mean more gradual evolution.

Understanding this allows us to predict the behavior of similar cosmic structures and their developmental timelines.

Dark Matter Halo

Dark matter halos represent an invisible component of galaxies that envelops the visible matter and affects their gravitational pull. These halos are believed to contain the majority of a galaxy’s mass, influencing how galaxies form, rotate, and interact with one another.
In the model of the proto-galactic nebula, considering a mass distribution similar to dark matter halos provides insights into:

• Gravitational forces at play during galaxy formation.
• The radial dependency of mass within the forming structures.
• Long-term stability and rotational dynamics of the galaxy.

By comparing the nebula's density distribution with that of a dark matter halo, astrophysicists can better understand the underlying structure and formation processes of galaxies.