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Snells Law is a method for calculating refraction. When things go from low refraction to high refraction, light angles down. When things go from high refraction to low refraction light angles up. This bends/changes the angle of light.
Snells Law disproves "far away" stars because light would eventually be bent after millions of miles. The probability is too high that even a 0.0000001 degree chance in angle would cause the light to miss us completely making consistent star trails impossible. Our atmos forms a density gradient. Things that are buoyant rise (like clouds and balloons). Helium and Hydrogen rise, Hydrogen above Helium so for example, when light goes from Hydrogen to Helium light bends up towards you.
Snell's law is given by the following formula: n<small>1</small>sin(θ<small>1</small>) = n<small>2</small>sin(θ<small>2</small>), where θ<small>1</small> is the angle made between the normal to the boundary and the medium with index of refraction n<small>1</small>. For light travelling in an interstellar perfect vacuum of space, considering any boundary intersected by the light ray's path, n<small>1</small> = n<small>2</small> = 1. Thus, sin(θ<small>1</small>) = sin(θ<small>2</small>), and taking the arcsine of both sides gives that θ<small>1</small> = θ<small>2</small>. In other words, the light ray would not bend in a vacuum. However, outer space is not said to be empty. To have a 0.000 000 1 bend, we can first look at the case where a light ray hits a boundary at a 45 degree angle coming from a vacuum and entering a near-vacuum. In this case,
θ<small>1</small> = 45deg,
θ<small>2</small> = 45deg - 0. 000 000 1deg,
n<small>1</small> = 1, and thus
n<small>2</small> = n<small>1</small>sin(θ<small>1</small>)/sin(θ<small>2</small>).
Using trigonometric identities and Taylor series approximations, to a large number of significant figures we have that sin(θ<small>2</small>) = 0.9999999*sqrt(2)/2. Since sin(45deg) = sqrt(2)/2, we have that n<small>2</small> = 1/0.9999999 = 1.000 000 1 to good approximation. Most gas in outer space is said to be hydrogen, which has a lower index of refraction than air at similar pressure and temperature, thus, if we use the Ciddor equation to estimate the gas density needed for n<small>2</small>, hydrogen gas will require a higher density to achieve the same index of refraction. For example, if we calculate (ignoring the scale of the numbers) using Ciddor that our outer space gas patch requires a density of 0.5atm to bend the light by the given angle, we will actually require a larger density such as 0.6atm. To get in the ballpark of the actual numbers, we can use the Ciddor equation, but we are well outside of the usual values of the input variables. We get the equation y = (3.37615288E-06)*x+0.99999992 for a trendline when fixing the temperature at -40degC and the relative humidity at 0, and letting x, the pressure in kPa vary, where y is the index of refraction. If y = 1.000 000 1, then the pressure is 0.0533151241658kPa. As mentioned, for hydrogen gas, this would be higher, but since this is a ballpark calculation, so we will not alter the required pressure. The pressure in outer space is said to be 1.322E-11 Pa, or 1.322E-14 kPa. We will just look at orders of magnitude. We needed a pocket of gas in space with a pressure of about 1E-2 kPa of pressure, but outer space is said to have a pressure of 1E-14 kPa, which is 10^12, or a trillion times smaller than it needs to be. This all assumes that just one pocket of gas in space causes the light to refract, and likely, a light ray would encounter multiple pockets of gas if any, however, the new direction for the light ray would likely be random, making it difficult for the light ray to accumulate any significant change in direction as it travelled. In summary, it seems unlikely for a light ray to encounter a dense enough patch of interstellar gas to cause a 0.000 000 1deg change in direction. More work will need to be done to determine if Snell's law predicts that the stars' light will miss us.
All chemical/elements have a refractive index so light will always change slightly everywhere it goes. these angular changes become extremely significant over many miles. When combining elements this can greatly exaggerate the refractivity. For example, by placing sugar in water you can see light bend using a laser pointer. Add to this diffusion, reflection and other important things that can happen to light in the atmos which require study.
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