Why is refractive index of atmosphere different at different altitudes

Variations in Refractive Index of atmosphere with altitude ?? The queries i have are related to refraction of light. 1. How does the refractive index of air vary with the altitude ? If possible explain the reason for change.

2. Does the amplitude of a wave change during refraction?

Refractive index increases from 1 in outer space to the refractive index of air at sea level. I'm trying to find an expression for the refractive index of air at a certain height, anyone?

Thanks

To a good approximation, the quantity n-1 is proportional to the density of air. So using a table of air density, and the fact that n-1 is 2.9x10-4 at sea level (density = 1.2 kg/m3), you can get a good idea of the refractive index at different altitudes. A table of air density vs. altitude is here: http://www.usatoday.com/weather/wstdatmo.htm More details, and a more accurate way to calculate n-1, is here:

http://www.kayelaby.npl.co.uk/general_physics/2_5/2_5_7.html

Note the formula after the words "For air at a temperature t °C and a pressure p Pa, ..." The expression shown is roughly proportional to p/tabsolute (with some minor corrective factors), which is proportional to density.

To a good approximation, the quantity n-1 is proportional to the density of air. So using a table of air density, and the fact that n-1 is 2.9x10-4 at sea level (density = 1.2 kg/m3), you can get a good idea of the refractive index at different altitudes. A table of air density vs. altitude is here: http://www.usatoday.com/weather/wstdatmo.htm More details, and a more accurate way to calculate n-1, is here:

http://www.kayelaby.npl.co.uk/general_physics/2_5/2_5_7.html

Note the formula after the words "For air at a temperature t °C and a pressure p Pa, ..." The expression shown is roughly proportional to p/tabsolute (with some minor corrective factors), which is proportional to density.

Thanks again.

You're welcome.

Note that n-1 is proportional to P/Tabsolute, not P. Assuming T is constant will skew the result.

Air pressure is proportional to density. I have found a model for air pressure: [tex]\text{p0} e^{-z/7000}[/tex] Where p0 is the air pressure at sea level (101.33 kpa), z is the height in metres. It works surprisingly well with the table you gave above. Then hence: [tex]\text{n = } 1+0.00029 e^{-z/7000}[/tex] I have made a plot in mathematica: [PLAIN]http://www.jamesandrews.eu/project/n_vs_height.jpeg [Broken] Note: x axis is height from sea level in metres, y is n (unitless) Does this sound reasonable?

Many thanks

Air pressure is proportional to density.

I have found a model for air pressure: [tex]\text{p0} e^{-z/7000}[/tex] Where p0 is the air pressure at sea level (101.33 kpa), z is the height in metres. It works surprisingly well with the table you gave above. Then hence: [tex]\text{n = } 1+0.00029 e^{-z/7000}[/tex] I have made a plot in mathematica: [PLAIN]http://www.jamesandrews.eu/project/n_vs_height.jpeg [Broken] Note: x axis is height from sea level in metres, y is n (unitless)

Does this sound reasonable?

Great, thanks for your help. Well I'm just going to use this model here, it does match up roughly with the data you provided earlier. I'm trying to calculate the time it takes for a signal to reach sea level, when emitted from a satellite 20,200km above sea level, with atmospheric effects. Velocity is given by v = c/n(z) where n is the refractive index at height z above sea level. v = (c/2.9/10^4)*Exp[z/7000] Now, plotting this gives: [PLAIN]http://www.jamesandrews.eu/project/v_vs_height.jpeg [Broken] Which seems pretty reasonable. How what I need is an expression to calculate the total time it takes for light to reach the earths surface, from a given height z. I realise it will have to be some integral, but after an hour or so I have run out of steam. Any help/pointers will be great.

Kind regards

[tex]\text{time} = \int {dt} = \int \frac{dx}{dx/dt} = \int \frac{dx}{v} = \int \frac{n \ dx}{c} \ \ \text{or} \ \ \frac{1}{c} \int {n \ dx} [/tex]

Thank you so much, you are a genius. I actually got pretty close to that ;)

I have thought about this some more. The calculation you want can be done without detailed knowledge of the index-vs.-altitude dependence, you just need to know the pressure at sea level (or wherever the light beam is to be "observed") as well as at the satellite's altitude. Note: fairly detailed solution follows; scroll down to the bottom for final result.

We want to know the time it takes for a light signal to reach a point on Earth at height h0 from a satellite located at altitude hs. The beam of light makes an angle θ from the vertical. If the beam is directed straight downward, then θ=0 and the cosθ factor that appears in the derivation is simply equal to 1 and may be ignored.

In other words, the total time is simple the time x/c that light would take if traveling through a vacuum, plus an additional time Δt. We'll now derive an expression for Δt which does not involve any integrals.

Using the (very good) approximation that (n-1) is proportional to the density of air, ρ, we have

Assume g is constant over altitude changes small compared to the 6400 km radius of the Earth.

We are getting there, but we need to know what the constant k is. As long as we know that the refractive index is nk at some known pressure and temperature Pk and Tk, then

If I have time tomorrow, I'll put some numbers on this. Note that if R uses N/m2 pressure units then mm should be expressed in kg/mole, instead of the more familiar g/mole values given in the periodic table.

Yes, some numbers would be useful to verify this. Using the previous method, I calculated that light was slowed down by (delta t) 3.33x10^-8 seconds. So I am assuming that for nk = 1.00029 (sea level), this would mean that Pk would be the average pressure at sea level, and Tk would be the average temperature at sea level? What would you use for Tk in this case? Obviously Tk varies quite a bit around the globe.

Kind regards

I'm trying to calculate the time it takes for a signal to reach sea level, when emitted from a satellite 20,200km above sea level, with atmospheric effects.

Using the previous method, I calculated that light was slowed down by (delta t) 3.33x10^8 seconds.

Much better now

At least there is no doubt it is a correct ballpark. 10-8 sec is the difference between light traveling 10 km in vacuum and in air at sea level.

Much better now

At least there is no doubt it is a correct ballpark. 10-8 sec is the difference between light traveling 10 km in vacuum and in air at sea level.

. . . [tex] \begin{flalign*} \Delta t & = & & \frac{k (P_0-P_s)}{c \ g \ \cos \theta} \\ \\ & = & & \frac{1}{\cos \theta} \cdot \frac{R}{c \ g \ m_m} \cdot \frac{T_k \ (n_k-1)}{P_k} \cdot (P_0-P_s) \end{flalign*} [/tex] And the total time is given by
Note: We are neglecting the deviation in θ due to refraction. We also neglect effects due to the Earths' curvature, which is okay as long as θ is not very close to 90o, i.e. the satellite does not appear close to the horizon when viewed from the observation point.

If I have time tomorrow, I'll put some numbers on this. Note that if R uses N/m2 pressure units then mm should be expressed in kg/mole, instead of the more familiar g/mole values given in the periodic table.

Yes, some numbers would be useful to verify this. . . . So I am assuming that for nk = 1.00029 (sea level), this would mean that Pk would be the average pressure at sea level, and Tk would be the average temperature at sea level? What would you use for Tk in this case? Obviously Tk varies quite a bit around the globe.

Kind regards

The quantity Tk*(nk-1)/Pk, which appears in the equation for Δt, is very nearly a constant over a wide range of temperatures and pressures, and is equal to 7.90*10-7K/Pa.

Using g=9.8 m/s2, and mm = 28.97 g/mol = 28.97*10-3 kg/mol for air, we have

R / (c g mm) = 9.8*10-8 s/K​

Δt = 7.7*10-14 sec/Pa * (P0-Ps) / cosθ​

Δt = 7.8 * 10-9 sec / cosθ​

Something about these answers does feel right to me, for several reasons: 1) the index of refraction varies with wavelength and air can scatter quite a lot of light. Both of these depend on the composition of the air, which varies with altitude (water vapor, ozone, CO2, and the sodium layer, for example).

http://en.wikipedia.org/wiki/Sodium_layer

2) temperature effects have not been considered, and the temperature varies wildly with altitude:

http://www.physicalgeography.net/fundamentals/7b.html

3) optical transmission through the atmosphere is highly complicated- there are several codes that are used to model the process (HITRAN and MODTRAN, for example).

http://www.modtran.org/features/index.html#isr

http://www.cfa.harvard.edu/HITRAN/

So the question really becomes one of needed accuracy: something left out of this discussion as well.

Also about the ionosphere, due to solar radiation ionising gasses in the ionosphere, there are free ions wondering around which can effect some wavelengths of magnetic radiation.

Now, data being sent from satellites don't emit light, they emit radio waves with wavelengths of ~1cm, how would this effect this solution? I assume that we would just have to adjust nk for this wavelength?

The quantity Tk*(nk-1)/Pk, which appears in the equation for Δt, is very nearly a constant over a wide range of temperatures and pressures, and is equal to 7.90*10-7K/Pa.

Also about the ionosphere, due to solar radiation ionising gasses in the ionosphere, there are free ions wondering around which can effect some wavelengths of magnetic radiation.

Now, data being sent from satellites don't emit light, they emit radio waves with wavelengths of ~1cm, how would this effect this solution? I assume that we would just have to adjust nk for this wavelength?

I don't know the details of how ions affect things, and don't know what a good reference would be for looking that up. Googling ionospheric refraction of radio waves does yield some info:

http://www.google.com/#sclient=psy&...=1&bav=on.2,or.r_gc.r_pw.&fp=44b3839bd44fe786