# Light Rays that are Perfectly Parallel

I just heard this simple reasoning in a documentary film:

Light rays from distant stars are perfectly parallel.

This is pretty interesting thought. In nature, it is hard to find something really perfect. For example, there are no isolated systems -- only quasi-isolated. But here it looks like indeed the light rays are perfectly parallel.

Maybe this can be confirmed by calculations? -- e.g. that angle between two light rays cannot be infinitely small (Planck constant will be probably involved?).

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In this case "perfectly parallel" means "more closely parallel than can be detected".

How parallel is that?

Consider the geometry. The biggest angular deviation possible is between a photon originating from the left side of the star impinging on the right side of the detector and vice versa.

By the small angle approximation we can write this difference as

$$\Delta \theta = 2\frac{(r_{star} + r_{scope})}{d_{star}}$$

where the $r$'s are the radii of the objects and the $d$ is the distance from the star to the Earth.

A nearby star is Alpha Centari A.

• Distance: 4.93 light years $\approx 4.67 \times 10^{16}\text{ m}$
• Diameter: 440,000 miles $\approx 7.08 \times 10^8\text{ m}$

A big telescope is either of the 10 meter jobs at the Keck observatory.

Which gives us a maximum angular deviation of

$$\Delta \theta_{max} \approx 2\frac{7.1 \times 10^8}{4.7 \times 10^{16}} \approx 1.5 \times 10^{-8} \text{ radians} \approx 2.7 \times 10^{-6} \text{degrees}$$

That's pretty parallel.

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Strictly speaking, light rays from distant stars are not perfectly parallel, but the typical angle (in radians) between them can be estimated as the diameter of the star divided by the distance from the star to the Earth; this value may equal $10^{-7}$ or less, so the rays are parallel to high accuracy.

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Actually, that is not true. You can easily calculate the angle by which rays from opposite sides of a star fail to be parallel; it's just the ratio of the diameter of the star to its distance. A typical value for a close Sun-like star would be something like $10^6\text {km}/100\text{ ly} = 10^{-9}$ (radians), which is just beyond the current limit of optical telescopes (0.001 arcsecond is about $5\times 10^{-9}$). One could say the rays are effectively parallel, though, since we are unable to resolve them separately.

If the math isn't enough to convince you, there are actually a few stars which can be directly photographed, either because they are unusually close or because they are unusually large. Of course, you can photograph any star, but it usually appears as just a point of light; however, if your telescope has good enough angular resolution to distinguish between the opposite sides of the star, you will instead get a blob, not a point. Examples of stars that have been imaged in this way include Betelgeuse (which is exceptionally large) and Gliese 229 B (which is relatively close).

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