Question

Current measurements and cosmological theories suggest that only about \(4 \%\) of the total mass of the universe is composed of ordinary matter. About \(22 \%\) of the mass is composed of dark matter, which does not emit or reflect light and can only be observed through its gravitational interaction with its surroundings (see Chapter 12). Suppose a galaxy with mass \(M_{\mathrm{G}}\) is moving in a straight line in the \(x\) -direction. After it interacts with an invisible clump of dark matter with mass \(M_{\mathrm{DM}}\), the galaxy moves with \(50 \%\) of its initial speed in a straight line in a direction that is rotated by an angle \(\theta\) from its initial velocity. Assume that initial and final velocities are given for positions where the galaxy is very far from the clump of dark matter, that the gravitational attraction can be neglected at those positions, and that the dark matter is initially at rest. Determine \(M_{\mathrm{DM}}\) in terms of \(M_{\mathrm{G}}, v_{0},\) and \(\theta\).

Short answer

Question: Determine the mass of the dark matter clump in terms of the mass of the galaxy, its initial speed, and the angle by which its velocity direction changes after the interaction. Answer: The mass of the dark matter clump is given by \(M_{\mathrm{DM}} = M_{\mathrm{G}}\sqrt{1 - 0.5 \cos\theta}\).

Step-by-step solution

Understand the principle of conservation of linear momentum

To solve this problem, we should use the principle of conservation of linear momentum. As there is no external force acting on the galaxy-dark matter system, the total momentum must be conserved. Before the interaction, the momentum is in the \(x\)-direction, and after the interaction, the momentum has a direction change. We can write the conservation of linear momentum equation for the \(x\) and \(y\) components separately: $$ M_{\mathrm{G}}v_0 = M_{\mathrm{G}}(0.5v_0 \cos\theta) + M_{\mathrm{DM}}v_{fx} $$ And, $$ 0 = M_{\mathrm{G}}(0.5v_0 \sin\theta) - M_{\mathrm{DM}}v_{fy} $$

Solve for the final velocities of the dark matter clump

Now, let's solve for \(v_{fx}\) and \(v_{fy}\) in terms of \(M_{\mathrm{G}}, v_0,\) and \(\theta\). From the first equation, we get: $$ v_{fx} = \frac{M_{\mathrm{G}}(v_0 - 0.5v_0 \cos\theta)}{M_{\mathrm{DM}}} $$ From the second equation, we get: $$ v_{fy} = \frac{M_{\mathrm{G}}(0.5v_0 \sin\theta)}{M_{\mathrm{DM}}} $$

Calculate the total momentum after the interaction

The total momentum of the galaxy-dark matter system after the interaction can be calculated using the square root of the sum of the squares of the \(x\) and \(y\) components: $$ p_f = \sqrt{(M_{\mathrm{G}}(0.5v_0 \cos\theta) + M_{\mathrm{DM}}v_{fx})^2 + (M_{\mathrm{G}}(0.5v_0 \sin\theta) - M_{\mathrm{DM}}v_{fy})^2} $$

Use the conservation of linear momentum

The initial momentum of the galaxy is given by \(M_{\mathrm{G}}v_0\). Since momentum is conserved, we have: $$ M_{\mathrm{G}}v_0 = p_f $$ Now, substitute the values for \(v_{fx}\) and \(v_{fy}\) from steps 2 and 3: $$ M_{\mathrm{G}}v_0 = \sqrt{(M_{\mathrm{G}}(0.5v_0 \cos\theta) + M_{\mathrm{DM}}\frac{M_{\mathrm{G}}(v_0 - 0.5v_0 \cos\theta)}{M_{\mathrm{DM}}})^2 + (M_{\mathrm{G}}(0.5v_0 \sin\theta) - M_{\mathrm{DM}}\frac{M_{\mathrm{G}}(0.5v_0 \sin\theta)}{M_{\mathrm{DM}}})^2} $$ Simplifying the expression: $$ (M_{\mathrm{G}}v_0)^2 = (M_{\mathrm{G}}v_0)^2 (1 - 0.5 \cos\theta)^2 + (M_{\mathrm{G}}(0.5v_0 \sin\theta))^2 $$

Solve for \(M_{\mathrm{DM}}\)

Now, we can solve for the mass of the dark matter clump \(M_{\mathrm{DM}}\): $$ 1 - 0.5 \cos\theta = \left(\frac{M_{\mathrm{DM}}}{M_{\mathrm{G}}}\right)^2 $$ Taking the square root of both sides: $$ \frac{M_{\mathrm{DM}}}{M_{\mathrm{G}}} = \sqrt{1 - 0.5 \cos\theta} $$ Finally, solving for \(M_{\mathrm{DM}}\): $$ M_{\mathrm{DM}} = M_{\mathrm{G}}\sqrt{1 - 0.5 \cos\theta} $$

Key concepts

Conservation of Linear Momentum

When we talk about the conservation of linear momentum, we're addressing one of the fundamental principles of physics that holds true in an isolated system where no external forces are acting. In essence, it states that the total momentum of a system remains constant if no external force is applied. It's much like saying when two dancers are spinning together and suddenly one of them lets go, they both continue spinning but in different directions and speeds - the total 'spin' remains unchanged.

In the context of our galaxy-and-dark matter interaction, we consider the galaxy and the dark matter clump as a closed system. Before the encounter, the galaxy is moving at a certain velocity, and the dark matter is at rest. Momentum is a vector quantity, so it has both magnitude (how fast something is moving) and direction. After the interaction, the direction and the distribution of this momentum change, but the total amount must stay the same because there are no external forces. This conservation is why we can set up equations to track how the momentum splits between the galaxy and the dark matter.

Using conservation laws like this is incredibly powerful because it allows us to uncover information about objects that we can't observe directly, such as dark matter. Since we can't see dark matter because it doesn't emit or reflect light, the way it interacts via gravity with objects we can see - like galaxies - provides vital clues to its characteristics, including its mass.

Gravitational Interaction

The gravitational interaction is a natural phenomenon by which all things with mass or energy are brought toward (or gravitate toward) one another. Gravity is responsible for the structure of the universe on the largest scales, from planets orbiting stars to the motions of galaxies and even clusters of galaxies.

In our example of the galaxy colliding with a clump of dark matter, although the dark matter cannot be seen, its presence is inferred from the gravitational effects on visible objects and light. You can think of this like feeling the wind on your face; you can't see the wind, but you can surely feel it. We know the dark matter is there because of how the galaxy moves after interacting with it.

Even though gravity is relatively weak compared to other fundamental forces, it dominates on the massive scales involved in astrophysics. These gravitational interactions become the main players in the cosmic dance that structures the universe, from the formation of galaxies to the trajectories of celestial bodies.

Cosmological Theories

The field of cosmology deals with the study of the origin, evolution, and eventual fate of the universe. It involves big questions and draws on the laws of physics and theoretical models to piece together the story of the cosmos. Contemporary cosmological theories suggest that the universe is made up of roughly 4% ordinary matter, 22% dark matter, and the remainder being dark energy. These components influence the universe's expansion rate, structure formation, and ultimate destiny.

Dark matter cannot be directly observed, yet it is imperative for cosmology because it contributes to the universe's overall mass-density causing a gravitational pull that affects the movements and formation of galaxies. Without the influence of dark matter, the galaxies we observe would spin themselves apart - the visible matter just doesn't provide enough gravity to hold them together.

To construct cosmological models, physicists rely on observational evidence such as the cosmic microwave background radiation, the distribution of galaxies, and the redshifts of distant objects. Through these observations, and assuming the conservation of momentum and the laws of thermodynamics, scientists develop theories that aim to make sense of the vast and mysterious cosmos.