The momentum of a photon is \(E / c .\) (a) Calculate the momentum per second delivered to the outer layers of a \(10^{2} \mathrm{L}_{\odot}\) star if all the photons are absorbed in that layer. (b) How does the force on the layer compare with the gravitational force on the layer if the layer has a radius \(R=100 R_{\odot}\) and a mass \(M=0.1 \mathrm{M}_{\odot}\) and the rest of the star has a mass of \(1 \mathrm{M}_{\odot}\)

Photon momentum refers to the momentum carried by a photon, which is a particle of light. Despite photons being massless, they still possess momentum, given by the equation: \( p = \frac{E}{c} \), where \( E \) is the energy of the photon and \( c \) is the speed of light. This property of photons is crucial in understanding how light can exert pressure, or force, on objects, including the outer layers of stars. This momentum transfer occurs when photons are absorbed by or reflected off a surface, imparting momentum to that surface. In the context of stellar physics, the absorption and emission of photons significantly impact a star's structure and behaviors by transferring energy and momentum. This is key in phenomena such as radiation pressure, which counteracts gravitational collapse in a star's lifecycle.

Stellar luminosity is a measure of the total amount of energy emitted by a star per unit of time. It's expressed in watts \( (W) \) or in terms of the Sun's luminosity \( (L_{\text{sun}}) \). In the exercise, the star's luminosity is given as \( 10^2 \) times that of the Sun. Here's how we calculate it:

• First, identify the Sun's luminosity, which is approximately \( L_{\text{sun}} = 3.8 \times 10^{26} \text{ W} \).
• Next, multiply this value by \( 10^2 \) to find the total energy output: \( L_{\text{star}} = 10^2 \times 3.8 \times 10^{26} \text{ W} = 3.8 \times 10^{28} \text{ W} \).

This total energy output signifies the star's brilliance and plays a role in the amount of photon momentum delivered to its outer layers. A brighter star emits more photons, thereby transferring more energy and momentum to its surroundings.

Gravitational force is the attractive force between two masses, governed by Newton's law of universal gravitation. This law states that any two objects with mass experience a force of attraction that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. The formula for gravitational force is given by: \( F_g = \frac{G \times M_1 \times M_2}{R^2} \), where:

• \( G \) is the gravitational constant, approximately \( 6.674 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2} \).
• \( M_1 \) and \( M_2 \) are the masses of the two objects.
• \( R \) is the distance between the centers of the two masses.

In the exercise, we apply this formula to find the gravitational force between the outer layer of the star (mass \( 0.1 M_{\text{sun}} \)) and the rest of the star (mass \( 1 M_{\text{sun}} \)) with a radius \( R = 100 R_{\text{sun}} \). By plugging in these values, we determine the gravitational force: \[ F_g = \frac{(6.674 \times 10^{-11}) \times (1.989 \times 10^{30}) \times (1.989 \times 10^{29})}{(6.96 \times 10^{10})^2} \text{ N} \] This helps us understand how strong the gravitational pull is in comparison to the photon momentum-derived force.

Newton's law of universal gravitation is a fundamental principle describing the attractive force between two masses. According to this law:

• Every point mass attracts every other point mass by a force acting along the line intersecting both points.
• The force is directly proportional to the product of the two masses.
• The force is inversely proportional to the square of the distance between the points.
• The formula is: \( F = G \frac{m_1 m_2}{r^2} \).

This law helps explain planetary motions, star formations, and various cosmic phenomena. In the exercise provided, it's used to calculate the gravitational force acting on the outer layer of a star. Plugging in the values for the constants and the given masses and radius: \[ F_g \text{ approximately equals } 5.48 \times 10^{22} \text{ N} \]. Comparing this to the photon momentum-derived force (\( 1.27 \times 10^{20} \text{ N} \)), we see that gravitational force dominates. This highlights the balancing act of forces within a star, maintaining its structure and dynamics.