<p>I found some interesting information <a href="http://www.arrl.org/files/file/Technology/pdf/119962.pdf" rel="nofollow noreferrer">in this vulgarization paper</a> by Ian Poole.</p>
<p>A first point is electron density in ionosphere changes between day and night, so the resulting bend will be different:
<a href="https://i.sstatic.net/oWsj3.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/oWsj3.png" alt="enter image description here" /></a></p>
<p>This <a href="http://www.astrosurf.com/luxorion/qsl-mars-communication.htm" rel="nofollow noreferrer">very interesting site explains</a> notably that there is a</p>
<blockquote>
<p><em>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</em></p>
</blockquote>
<p>The article includes an illustration of the transition from angular deflection to complete reflection as a function of angle (click for full size):</p>
<p><a href="https://i.sstatic.net/z4H7O.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/z4H7Om.jpg" alt="enter image description here" /></a></p>
<p><strong>caption:</strong> 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.</p>
<p>But the best site I found on the subject <a href="http://www.cv.nrao.edu/course/astr534/Introradastro.html" rel="nofollow noreferrer"><strong>is that one.</strong></a> It states that</p>
<blockquote>
<p>Ionospheric reflection (not absorption) prevents photons with wavelengths &gt; 30 m (f&lt; 10 MHz) to reach the ground
[...]</p>
<p>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.</p>
</blockquote>
<p>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.</p>

CC BY-SA 3.0
2017-08-29T14:36:49.197Z
Those are some great illustrations! Figure 5 is noteworthy, because it shows that for transmitted rays, they return approximately to their original direction once they leave the ionized regions. This is *different* than for refraction by air, because once the ray enters air, it *says in air*. But the ionized layers are shells. There will be an effect because the shells are curved and not slabs, but now I realize it will be a much smaller effect than I thought. OK I'll take a closer look at your links. Thanks!
uhoh
CC BY-SA 3.0
2017-08-29T14:37:38.383Z
Refraction by a slab with parallel faces does not change the direction of a ray: https://i.sstatic.net/k0QK4.png and also https://i.sstatic.net/ZMrgu.png
uhoh
CC BY-SA 3.0
2017-09-03T03:54:42.297Z
This is a helpful answer. HF waves experience much stronger effects than VHF or UHF. There is plenty of short-wave radio and amateur radio experience here, but most radio astronomy is done at much higher frequency where the ionospheric deflection is much smaller. This is due in part to the science available at those wavelengths, and in part due to the better angular resolution a given instrument has as the wavelength decreases. I added an image from one of your links, it's only an illustration but it helps illustrate that the effects are larger at lower frequency.
uhoh
CC BY-SA 3.0
2017-09-03T03:57:05.730Z
I won't accept an answer yet, still holding off for an answer that has addresses the angular deflection quantitatively. But here's the bounty. I may add another later if necessary.
uhoh
CC BY-SA 4.0
2020-09-21T07:15:11.990Z
This does not answer the question. In fact, the implication is that effects are much larger in radio astronomy, but they are not.
ProfRob
CC BY-SA 4.0
2020-09-21T16:31:57.190Z
At the frequencies where most radio astronomy is done anyway.
ProfRob
CC BY-SA 3.0
2017-08-29T14:02:53.480Z
2017-09-03T03:50:35.763Z
2020-06-17T09:47:40.653Z
J. Chomel
<p>The refractive deviations in position are very similar for both radio and optical astronomy, until you consider very low frequency radio waves (<span class="math-container">$&lt;200$</span> MHz) when the effect becomes rapidly larger.</p>
<p>For plane parallel refraction an approximation for the deviation you are talking about is
<span class="math-container">$$\Delta \theta \simeq (n-1) \cot \theta,$$</span>
where <span class="math-container">$\theta$</span> is the observed elevation, <span class="math-container">$\Delta \theta$</span> is the change in elevation from its true value due to refraction and <span class="math-container">$n$</span> is the refractive index averaged over airmass.</p>
<p>According to <a href="http://www.gb.nrao.edu/%7Ermaddale/Education/AtmosphereWeatherForRadioAstronomy.pdf" rel="nofollow noreferrer">this source</a> from the Green bank radio telescope, they use something like this, with an added model for how <span class="math-container">$n$</span> varies with height, scaled by the atmospheric pressure. The largest value of <span class="math-container">$n$</span> 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.</p>
<p>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.</p>
<p>Here is <a href="https://www.fig.net/resources/proceedings/fig_proceedings/fig_2002/Js28/JS28_rueger.pdf" rel="nofollow noreferrer">another source</a> that gives some algorithms to calculate the effective (small) real refractive index for radio waves, with similar results.</p>
<p>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.</p>
<p>I did manage to find something about positional refraction at these low frequencies. The <a href="https://en.wikipedia.org/wiki/Low-Frequency_Array_(LOFAR)" rel="nofollow noreferrer">LOFAR</a> 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 <a href="http://lwa.phys.unm.edu/chicago08/talks/anderson_software.pdf" rel="nofollow noreferrer">this presentation</a>, which contains a section on refraction and in particular, the figure below.</p>
<p><a href="https://i.sstatic.net/4auFC.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/4auFC.png" alt="Refraction at low frequencies" /></a></p>
<p>Thus for low frequency radio astronomy (&lt;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 <em>degrees</em> at 50MHz and 30MHz respectively.</p>

CC BY-SA 3.0
2017-08-30T09:27:57.440Z
For a slab (goes in and then goes out again, which is the case for the ionosphere), the deviation has to be zero. Is that equation for one interface only, like from space to air for visible wavelengths? The ionospheric layers are more like shells, with no charged particles both above and below, so I don't think that equation applies - if it is for planar surfaces.
uhoh
CC BY-SA 3.0
2017-08-30T10:08:27.973Z
@uhoh The equation is the approximation for the deviation of a radio source from its expected position due to the atmosphere above the observer.
ProfRob
CC BY-SA 3.0
2017-08-30T11:42:37.130Z
I see, this is for any EM wave and a simple dielectric, not the interaction with the plasma in the ionosphere. OK I'll take a look, thanks!
uhoh
CC BY-SA 3.0
2017-08-30T12:12:23.863Z
@uhoh The radio waves have to pass through everything above the Earth's surface. Only the real part of the refractive index changes the phase velocity and causes refraction. This appears to be very small.
ProfRob
CC BY-SA 3.0
2017-09-01T11:45:29.330Z
The refraction can be so strong that lower frequencies transmitted from Earth are bent back to the surface; "skip". I am pretty sure that at those frequencies, incoming waves from space would be reflected back into space - there are no one-way mirrors. I think your "very small" only applies to higher frequencies, and the lowest frequencies really do experience substantial effects. I may have to read up and answer my own question. I wish there was a way to "[escalate to an expert](https://astronomy.meta.stackexchange.com/q/376/7982)" in this case a low-frequency observational radio astronomer.
uhoh
CC BY-SA 3.0
2017-09-01T16:20:37.130Z
@uhoh Then please change your question to ask exactly what you want to know. I have given you the answer that is used at the Green Bank radio telescope.
ProfRob
CC BY-SA 3.0
2017-09-02T03:09:07.027Z
According to Wikipedia Green Bank's frequency coverage is  290 MHz to 100 GHz, and I'm guessing most of the work is not done at 290 MHz, so the selection of this instrument biases your otherwise excellent answer. Check http://www.ncra.tifr.res.in/ncra/gmrt/gmrt-users/observing-help-for-gmrt-users/low-frequency-radio-astronomy/ch16.pdf and http://www.astron.nl/eris2013/Documents/11_pizzo_low_freq_interferometry.pdf and https://www.mpifr-bonn.mpg.de/1040895/deBruyn_LowFreq_red.pdf for example. There is interest in putting an LF array in space to get above the effects of the ionosphere.
uhoh
CC BY-SA 4.0
2020-04-24T01:18:35.940Z
The first link is was moved here: http://www.ncra.tifr.res.in/ncra/gmrt/gmrt-users/low-frequency-radio-astronomy/ch16.pdf
uhoh
CC BY-SA 4.0
2020-09-22T17:25:52.470Z
Bravo! Thank you! :-)
uhoh
CC BY-SA 4.0
2017-08-30T08:02:29.810Z
2020-09-22T06:54:36.450Z
2020-09-22T06:54:36.450Z
ProfRob
<p>For atmospheric refraction of visible light, <a href="https://en.wikipedia.org/wiki/Atmospheric_refraction#Values" rel="nofollow noreferrer">Wikipedia gives</a> 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.</p>

<p>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.</p>

<p>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.</p>

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

<hr>

<p>I started thinking about this after asking <a href="https://space.stackexchange.com/q/22724/12102">How many stations could one hear with an AM/FM radio in front of the ISS' cupola window?</a> which includes the image below.</p>

<p><strong>below:</strong> from the Radio Jove Project's exercise <a href="https://radiojove.gsfc.nasa.gov/education/educ/radio/tran-rec/exerc/iono.htm" rel="nofollow noreferrer">The Effects of Earth's Upper Atmosphere on Radio Signals</a>.</p>

<p><a href="https://i.sstatic.net/r3nps.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/r3nps.gif" alt="enter image description here"></a></p>

CC BY-SA 3.0
2017-08-21T13:11:14.513Z
If you track down tables of  `n` (index of refraction) at radio wavelengths, your problem is solved.  Your diagrams appear to show purely reflective effects.
Carl Witthoft
CC BY-SA 3.0
2017-08-21T15:30:44.247Z
@CarlWitthoft The diagram is obviously a cartoon, not to scale, and not to be taken too literally. However the process it illustrates is indeed *refraction, and not reflection*. Refraction by the plasma of the ionosphere is a complex problem and not something you can just look up in tables. In fact the index is literally a complex number `n+jk`.
uhoh
CC BY-SA 3.0
2017-08-22T12:29:12.700Z
Nothing wrong with complex  `n = nx + i*ny`
Carl Witthoft
CC BY-SA 3.0
2017-09-05T06:33:47.077Z
Do a search on "LOFAR position refraction", I think you will find useful material.
ProfRob
CC BY-SA 3.0
2017-09-05T06:49:16.217Z
@RobJeffries omg what a nightmare, no wonder there is (at least some) interest in putting a low frequency array in space. On [slide 29 here](http://www.phys.unm.edu/~lwa/chicago08/talks/anderson_software.pdf) I see > 1 degree for 20-30 MHz. At some point, do you think you might augment your current answer with a mention of the the more exotic LF regime and ionosphere? There really is no maximum or answer for "how large" but the transition from arcseconds in multi-GHz to degrees at tens of MHz is pretty amazing.
uhoh
CC BY-SA 3.0
2017-09-05T07:01:12.413Z
@RobJeffries also thank you for pointing me towards LOFAR. It's "different" astronomy and interesting to read about, and as an intermittent ham radio operator I like to look at all those antennas!
uhoh
CC BY-SA 4.0
2020-09-21T03:25:00.643Z
**Last chance** for somebody to post a quantitative answer! Comments contain enough information to write one.
uhoh
<radio-astronomy><atmospheric-effects><adaptive-optics>
How large does refraction become in radioastronomy?
uhoh
# How large does refraction become in radioastronomy?

(Asked by: username_0 on 2017-08-21)

For atmospheric refraction of visible light, [Wikipedia gives](https://en.wikipedia.org/wiki/Atmospheric_refraction#Values) 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?

---

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?](https://space.stackexchange.com/q/22724/12102) which includes the image below.

**below:** from the Radio Jove Project's exercise [The Effects of Earth's Upper Atmosphere on Radio Signals](https://radiojove.gsfc.nasa.gov/education/educ/radio/tran-rec/exerc/iono.htm).

[![enter image description here](https://i.sstatic.net/r3nps.gif)](https://i.sstatic.net/r3nps.gif)

**Tags:** radio-astronomy, atmospheric-effects, adaptive-optics
> **Comments:**
> **username_1**: If you track down tables of  `n` (index of refraction) at radio wavelengths, your problem is solved.  Your diagrams appear to show purely reflective effects.  
> **username_0**: @CarlWitthoft The diagram is obviously a cartoon, not to scale, and not to be taken too literally. However the process it illustrates is indeed *refraction, and not reflection*. Refraction by the plasma of the ionosphere is a complex problem and not something you can just look up in tables. In fact the index is literally a complex number `n+jk`.  
> **username_1**: Nothing wrong with complex  `n = nx + i*ny`  
> **username_2**: Do a search on "LOFAR position refraction", I think you will find useful material.  
> **username_0**: @RobJeffries omg what a nightmare, no wonder there is (at least some) interest in putting a low frequency array in space. On [slide 29 here](http://www.phys.unm.edu/~lwa/chicago08/talks/anderson_software.pdf) I see > 1 degree for 20-30 MHz. At some point, do you think you might augment your current answer with a mention of the the more exotic LF regime and ionosphere? There really is no maximum or answer for "how large" but the transition from arcseconds in multi-GHz to degrees at tens of MHz is pretty amazing.  
> **username_0**: @RobJeffries also thank you for pointing me towards LOFAR. It's "different" astronomy and interesting to read about, and as an intermittent ham radio operator I like to look at all those antennas!  
> **username_0**: **Last chance** for somebody to post a quantitative answer! Comments contain enough information to write one.  

---

## 2 answers

### Answer by username_2 on 2017-08-30. Score: 5 | Accepted answer

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](http://www.gb.nrao.edu/%7Ermaddale/Education/AtmosphereWeatherForRadioAstronomy.pdf) 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](https://www.fig.net/resources/proceedings/fig_proceedings/fig_2002/Js28/JS28_rueger.pdf) 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](https://en.wikipedia.org/wiki/Low-Frequency_Array_(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](http://lwa.phys.unm.edu/chicago08/talks/anderson_software.pdf), which contains a section on refraction and in particular, the figure below.

[![Refraction at low frequencies](https://i.sstatic.net/4auFC.png)](https://i.sstatic.net/4auFC.png)

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.

> **Comments:**
> **username_0**: For a slab (goes in and then goes out again, which is the case for the ionosphere), the deviation has to be zero. Is that equation for one interface only, like from space to air for visible wavelengths? The ionospheric layers are more like shells, with no charged particles both above and below, so I don't think that equation applies - if it is for planar surfaces.  
> **username_2**: @uhoh The equation is the approximation for the deviation of a radio source from its expected position due to the atmosphere above the observer.  
> **username_0**: I see, this is for any EM wave and a simple dielectric, not the interaction with the plasma in the ionosphere. OK I'll take a look, thanks!  
> **username_2**: @uhoh The radio waves have to pass through everything above the Earth's surface. Only the real part of the refractive index changes the phase velocity and causes refraction. This appears to be very small.  
> **username_0**: The refraction can be so strong that lower frequencies transmitted from Earth are bent back to the surface; "skip". I am pretty sure that at those frequencies, incoming waves from space would be reflected back into space - there are no one-way mirrors. I think your "very small" only applies to higher frequencies, and the lowest frequencies really do experience substantial effects. I may have to read up and answer my own question. I wish there was a way to "[escalate to an expert](https://astronomy.meta.stackexchange.com/q/376/7982)" in this case a low-frequency observational radio astronomer.  
> **username_2**: @uhoh Then please change your question to ask exactly what you want to know. I have given you the answer that is used at the Green Bank radio telescope.  
> **username_0**: According to Wikipedia Green Bank's frequency coverage is  290 MHz to 100 GHz, and I'm guessing most of the work is not done at 290 MHz, so the selection of this instrument biases your otherwise excellent answer. Check http://www.ncra.tifr.res.in/ncra/gmrt/gmrt-users/observing-help-for-gmrt-users/low-frequency-radio-astronomy/ch16.pdf and http://www.astron.nl/eris2013/Documents/11_pizzo_low_freq_interferometry.pdf and https://www.mpifr-bonn.mpg.de/1040895/deBruyn_LowFreq_red.pdf for example. There is interest in putting an LF array in space to get above the effects of the ionosphere.  
> **username_0**: The first link is was moved here: http://www.ncra.tifr.res.in/ncra/gmrt/gmrt-users/low-frequency-radio-astronomy/ch16.pdf  
> **username_0**: Bravo! Thank you! :-)  

---

### Answer by username_3 on 2017-08-29. Score: 2

I found some interesting information [in this vulgarization paper](http://www.arrl.org/files/file/Technology/pdf/119962.pdf) by Ian Poole.

A first point is electron density in ionosphere changes between day and night, so the resulting bend will be different:
[![enter image description here](https://i.sstatic.net/oWsj3.png)](https://i.sstatic.net/oWsj3.png)

This [very interesting site explains](http://www.astrosurf.com/luxorion/qsl-mars-communication.htm) 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):

[![enter image description here](https://i.sstatic.net/z4H7Om.jpg)](https://i.sstatic.net/z4H7O.jpg)

**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.**](http://www.cv.nrao.edu/course/astr534/Introradastro.html) 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.

> **Comments:**
> **username_0**: Those are some great illustrations! Figure 5 is noteworthy, because it shows that for transmitted rays, they return approximately to their original direction once they leave the ionized regions. This is *different* than for refraction by air, because once the ray enters air, it *says in air*. But the ionized layers are shells. There will be an effect because the shells are curved and not slabs, but now I realize it will be a much smaller effect than I thought. OK I'll take a closer look at your links. Thanks!  
> **username_0**: Refraction by a slab with parallel faces does not change the direction of a ray: https://i.sstatic.net/k0QK4.png and also https://i.sstatic.net/ZMrgu.png  
> **username_0**: This is a helpful answer. HF waves experience much stronger effects than VHF or UHF. There is plenty of short-wave radio and amateur radio experience here, but most radio astronomy is done at much higher frequency where the ionospheric deflection is much smaller. This is due in part to the science available at those wavelengths, and in part due to the better angular resolution a given instrument has as the wavelength decreases. I added an image from one of your links, it's only an illustration but it helps illustrate that the effects are larger at lower frequency.  
> **username_0**: I won't accept an answer yet, still holding off for an answer that has addresses the angular deflection quantitatively. But here's the bounty. I may add another later if necessary.  
> **username_2**: This does not answer the question. In fact, the implication is that effects are much larger in radio astronomy, but they are not.  
> **username_2**: At the frequencies where most radio astronomy is done anyway.  

---

