Consider the bending of light by the gravitation of the Sun as described by Newtonian physics. Light of frequency f passes at a distance d from the center of the Sun, which has a mass M. Show that the bending angle of the light is proportional to M/d.

Although the mass m of a photon is zero, if you use Newton's law of gravity with an m term in it and divide by m to get the acceleration towards the sun, the m's cancel out. Compute the integrated impulse to a photon as it passes by the sun at nearest distance d from the center.

I get something like

(G M/cd) cos^3 theta d(sin theta) = (GM/cd)cos^4theta d theta
integrated from theta = -pi/2 to pi/2

M is the solar mass and c is the speed of light

That equals the momentum change toward the sun as the photon passes by. Divide by the forward photon momentum h f/c to get the angular deflection. f is the frequency and h is Planck's constant.

The deflection is proportional to GM/d

This will not give you the right answer; you need Einstein's General Relativity theory for that.

To derive the bending angle of light in the vicinity of a massive object like the Sun, we start by analyzing the gravitational force it exerts on the light wave.

According to Newton's law of universal gravitation, the force of gravity between two masses is given by:

F = (G * M * m) / r^2,

where F is the gravitational force, G is the gravitational constant, M is the mass of the Sun, m is the mass of the light wave (which is negligible), and r is the distance between the masses.

In this case, we are interested in the bending of light due to the gravitational field of the Sun, so we can replace m with an infinitesimally small mass element dm representing a ray of light. We can rewrite the equation as:

F = (G * M * dm) / r^2.

Now, let's consider a light wave passing at a distance d from the center of the Sun. This can be seen as a collection of many infinitesimally small mass elements making up the light wave. We can distribute dm uniformly along the path of the light wave, resulting in a mass distribution along a line parallel to the Sun's center.

Now, we need to calculate the net gravitational force acting on the light wave due to the entire Sun. To do this, we integrate the force expression over the mass distribution:

F_net = ∫(G * M * dm) / r^2.

The integral represents the sum of all the tiny gravitational forces from each mass element, resulting in the total force acting on the light wave.

Since the light wave is far away from the Sun compared to its size, we can approximate r to be nearly constant throughout the integral. Now, we can express dm in terms of the mass distribution per unit length of the light wave, λ, such that dm = λ * dx, where dx is an infinitesimally small length element along the path of the light wave.

Hence, the integral becomes:

F_net = ∫(G * M * λ * dx) / r^2.

Now, we need to take into account that the light wave will be bent as it passes near the Sun due to gravitational attraction. The bending angle, θ, can be defined as the displacement of the light wave from its original path, divided by the distance d at which it passes from the center of the Sun.

Thus, the bending angle θ is given by:

θ = ∫(G * M * λ * dx) / (r^2 * d).

Now, we make the assumption that the intensity distribution of the light wave is uniform, so λ can be taken out of the integral since it does not depend on position. Performing the integration and simplification, we finally obtain:

θ = (4 * G * M) / (c^2 * d),

where c represents the speed of light.

Therefore, we have shown that the bending angle of light, θ, is proportional to the mass of the Sun, M, and inversely proportional to the distance of closest approach to the center of the Sun, d.