How large does refraction become in radioastronomy?

Question

For atmospheric refraction of visible light, Wikipedia gives the order of 1 arc minute at 45° altitude above the horizon, and 5.3 arc minutes at 10°. This is caused by the dielectric polarizability of all of the bound electrons in all the atoms of the atmosphere.

At the much lower HF frequencies of radio, the free electrons and ions will contribute, and some forms of radio communication have relied on refraction at large incident angles to deflect terrestrial signals back to the Earth at a distant ground station.

So I expect that at the lower frequencies used in radio astronomy, corrections to the observed location of radio sources due to ionospheric refraction could be much larger than those at visible wavelengths, but I am not sure.

How large does can this effect ever get? At what frequency? Are there ever corrections as large as 1 degree?

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I started thinking about this after asking How many stations could one hear with an AM/FM radio in front of the ISS' cupola window? which includes the image below.

below: from the Radio Jove Project's exercise The Effects of Earth's Upper Atmosphere on Radio Signals.

Answer 1

The refractive deviations in position are very similar for both radio and optical astronomy, until you consider very low frequency radio waves ($<200$ MHz) when the effect becomes rapidly larger.

For plane parallel refraction an approximation for the deviation you are talking about is $$\Delta \theta \simeq (n-1) \cot \theta,$$ where $\theta$ is the observed elevation, $\Delta \theta$ is the change in elevation from its true value due to refraction and $n$ is the refractive index averaged over airmass.

According to this source from the Green bank radio telescope, they use something like this, with an added model for how $n$ varies with height, scaled by the atmospheric pressure. The largest value of $n$ quoted is 1.00031 at ground level. This is basically the same as the refractive index of air at visible wavelengths and amounts to about 60 arcseconds at elevations of 45 degrees.

So, to my surprise, the effects of refraction on radio telescope pointing are similar to those for optical telescopes. It simply turns out that the real part of the refractive index (that controls the phase velocity of light and hence refraction) is just as close to 1 for radiowaves as it is for visible light.

Here is another source that gives some algorithms to calculate the effective (small) real refractive index for radio waves, with similar results.

This source claims the calculations are valid for mm-wavelengths and upwards. Of course they cannot be valid as one approaches the ionospheric cut-off at around 40 MHz (wavelengths of metres), where the refractive index will depart sharply from unity and the deflections must get commensurately larger.

I did manage to find something about positional refraction at these low frequencies. The LOFAR radio array can apparently do work down to frequencies as low as 10MHz, but the practical limit may be a little higher. Anyway, they do need to account for refraction in the ionosphere and I found this presentation, which contains a section on refraction and in particular, the figure below.

Thus for low frequency radio astronomy (<200 MHz), refraction is certainly a bigger effect than in the optical. e.g. At elevations of 45 degrees, the refracted positions are shifted by about 0.1 and 0.4 degrees at 50MHz and 30MHz respectively.

Answer 2

I found some interesting information in this vulgarization paper by Ian Poole.

A first point is electron density in ionosphere changes between day and night, so the resulting bend will be different:

This very interesting site explains notably that there is a

cut-off frequency for the ionosphere beyond which it loses its capacity to reflect shortwaves. Depending the latitude, the season and the solar activity mainly, during the day this frequency is around 3-10 MHz and goes down to about 2-6 MHz during the night

The article includes an illustration of the transition from angular deflection to complete reflection as a function of angle (click for full size):

caption: Space wave, ground wave, and ionospheric waves. Above a critical angle, waves escape in free space while waves emitted under a low incidence angle can reach very far countries. This is valid between approx. 1-500 MHz.

But the best site I found on the subject is that one. It states that

Ionospheric reflection (not absorption) prevents photons with wavelengths > 30 m (f< 10 MHz) to reach the ground [...]

Total internal reflection in the ionosphere at longer wavelengths makes the Earth look like a silvery ball from space, like the glass face of an underwater wristwatch viewed obliquely.

It goes on saying the atmosphere is not perfectly transparent at any radio frequency. And moreover it adds noise. It explains why the best sites for radio observation at higher frequencies are exceptionally high and dry.