Help Deriving formula for Einstein Radius

In summary, the Einstein ring radius can be calculated using the formula R_e = 2[GMx(L-x)/(Lc^2)]^1/2, where L is the distance to the source and x is the distance to the lens mass. This can be derived using the light bending formula for small angles and assuming Euclidean geometry. The source, lens, and observer must be lined up for an Einstein ring to form. Further assistance is available if needed.
  • #1
adwbizi1
1
0
I have been trying to show that the Einstein ring radius R_e = 2[GMx(L-x)/(Lc^2)]^1/2
to no avail. can someone who knows this show me, or at least point out the direction. I have a strong hunch that i'll have to use the light bending formula for small angles 
delta_phi = 4GM/(bc^2), and geometry.

The Einstein radius is the radius of the ring image formed when a bright source is exactly behind a spherically symmetric lens mass. L is the distance to the source, and x is the distance to the lens mass.
 
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  • #2
adwbizi1 said:
I have been trying to show that the Einstein ring radius R_e = 2[GMx(L-x)/(Lc^2)]^1/2
to no avail. can someone who knows this show me, or at least point out the direction. I have a strong hunch that i'll have to use the light bending formula for small angles
delta_phi = 4GM/(bc^2), and geometry.

The Einstein radius is the radius of the ring image formed when a bright source is exactly behind a spherically symmetric lens mass. L is the distance to the source, and x is the distance to the lens mass.

The formula on http://en.wikipedia.org/wiki/Einstein_ring" can be derived using the light bending formula for small angles. Also, assume Euclidean geometry and use [itex]\theta \doteq \sin\theta \doteq \tan\theta[/itex]. Click on the diagram, and take [itex]\theta_S = 0[/itex], since, for an Einstein ring, the source, lens, and observer have to be lined up.

If you need more help, just ask.
 
Last edited by a moderator:
  • #3
The formula you are trying to derive is correct, but here is a step-by-step explanation of how it can be derived using the light bending formula and geometry:

1. Start with the light bending formula for small angles:

Δφ = 4GM/(bc^2)

Where Δφ is the deflection angle, G is the gravitational constant, M is the mass of the lens, b is the impact parameter (perpendicular distance from the source to the lens), and c is the speed of light.

2. Rearrange the formula to solve for the impact parameter b:

b = 4GM/(c^2Δφ)

3. Substitute the value of Δφ for the deflection angle at the Einstein radius, which is equal to half of the total deflection angle for a spherically symmetric lens:

Δφ = θ_e/2

Where θ_e is the Einstein angle.

4. Substitute the value of the impact parameter b into the formula for the Einstein angle:

θ_e = 4GM/(c^2b)

5. Now, we need to find the value of the impact parameter b at the Einstein radius. This can be done by considering the geometry of the situation. At the Einstein radius, the source, lens, and observer form a right triangle, with the lens at the right angle. The distance from the observer to the lens is x, and the distance from the lens to the source is L-x. Using the Pythagorean theorem, we can find the value of the impact parameter b:

b^2 = x^2 + (L-x)^2

6. Substitute the value of b^2 into the formula for the Einstein angle:

θ_e = 4GM/(c^2[x^2 + (L-x)^2])

7. Simplify the formula by combining like terms and taking the square root on both sides:

θ_e = 2[GMx(L-x)/(Lc^2)]^1/2

And there you have it, the formula for the Einstein radius. I hope this helps and points you in the right direction for deriving it yourself. Keep in mind that this is just one way of deriving the formula, and there may be other approaches as well.
 

Related to Help Deriving formula for Einstein Radius

1. What is the Einstein radius?

The Einstein radius, also known as the Einstein ring, is a phenomenon in gravitational lensing where the light from a distant object is bent by the gravitational pull of a massive object in between, resulting in a ring-like image of the distant object.

2. How is the Einstein radius calculated?

The formula for calculating the Einstein radius is given by θE = √(4GM/c2 * dLS/dOS * (dOL/dOS - 1)), where G is the gravitational constant, M is the mass of the lensing object, c is the speed of light, and dLS, dOL, and dOS represent the distances between the lens, observer, and source, respectively.

3. What does the Einstein radius tell us about the lensing object?

The Einstein radius can provide information about the mass of the lensing object, as well as its distance from the observer and the source. It can also help determine the distribution of matter within the lensing object.

4. Can the Einstein radius be observed?

Yes, the Einstein radius has been observed in various astronomical objects, such as galaxies and galaxy clusters. It can be observed through telescopes or other imaging techniques that can detect the bending of light caused by gravitational lensing.

5. Are there any real-life applications of the Einstein radius?

The study of the Einstein radius and gravitational lensing has numerous applications in astrophysics and cosmology. It can help in the detection and study of dark matter, as well as provide insights into the formation and evolution of galaxies and galaxy clusters. It can also be used to measure the expansion rate of the universe and study the effects of dark energy.

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