<blockquote>
<p>...since we always observe the source on the axes of the dishes...</p>
</blockquote>
<p><em>Optical axis</em>, often. Mechanical/structural axis, <em><strong>not so fast!</strong></em></p>
<p>Radio astronomy is full of large antennas that are physically static, but electronically steered, and there are some fixed dishes where only the secondary reflector moves.</p>
<p>All of those will present a <span class="math-container">$\cos \theta$</span> like cross-section to the source.</p>
<p><strong>For 100% static antennas</strong> a recent example is <a href="https://en.wikipedia.org/wiki/Canadian_Hydrogen_Intensity_Mapping_Experiment" rel="nofollow noreferrer">CHIME</a> (cf. <a href="https://astronomy.stackexchange.com/q/28615/7982">How does Canada&#39;s radiotelescope CHIME manage polarization-dependent apodization of the fast f/0.25 optics?</a>)</p>
<p><a href="https://i.sstatic.net/1cVDT.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/1cVDT.jpg" alt="CHIME" /></a><a href="https://x.com/SKAO/status/1065277565429010432?mx=2" rel="nofollow noreferrer">source</a></p>
<p>and a famous, historic older and longer wavelength example is the <a href="https://en.wikipedia.org/wiki/Interplanetary_Scintillation_Array" rel="nofollow noreferrer">Interplanetary Scintillation Array</a> where the first pulsar was discovered by then graduate student <a href="https://en.wikipedia.org/wiki/Jocelyn_Bell_Burnell" rel="nofollow noreferrer">Jocelyn Bell Burnell</a> through careful and painstaking review of chart recorder data. (cf. <a href="https://astronomy.stackexchange.com/q/34532/7982">What exactly is interplanetary scintillation; what was the Interplanetary Scintillation Array looking for? Did it successfully observe any?</a>)</p>
<p><a href="https://i.sstatic.net/TplAAMrJ.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/TplAAMrJ.jpg" alt="Interplanetary Scintillation Array" /></a>(<a href="https://commons.wikimedia.org/wiki/File:Cmglee_Cambridge_MRAO_IPS_Array.jpg" rel="nofollow noreferrer">source</a>)</p>
<p>and the <a href="https://en.wikipedia.org/wiki/Ohio_State_University_Radio_Observatory" rel="nofollow noreferrer">Ohio State University Radio Observatory</a> of <a href="https://en.wikipedia.org/wiki/Wow!_signal" rel="nofollow noreferrer">Wow! signal</a> fame (cf. <a href="https://astronomy.stackexchange.com/q/35091/7982">What was the field of view of the Ohio State University Radio Observatory of Wow! signal fame?</a> and <a href="https://astronomy.stackexchange.com/q/35089/7982">Did comets 266P/Christensen or P/2008 Y2 (Gibbs) cause the Wow! signal?</a>)</p>
<p><a href="https://i.sstatic.net/OlkSVdI1.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/OlkSVdI1.jpg" alt="enter image description here" /></a>(<a href="https://commons.wikimedia.org/wiki/File:%D0%A0%D0%B0%D0%B4%D0%B8%D0%BE%D1%82%D0%B5%D0%BB%D0%B5%D1%81%D0%BA%D0%BE%D0%BF_%D0%A3%D0%BD%D0%B8%D0%B2%D0%B5%D1%80%D1%81%D0%B8%D1%82%D0%B5%D1%82%D0%B0_%D0%9E%D0%B3%D0%B0%D0%B9%D0%BE.jpg" rel="nofollow noreferrer">source</a>)</p>
<p>For static dishes, look no further than the late great <a href="https://en.wikipedia.org/wiki/Arecibo_Observatory" rel="nofollow noreferrer">Arecebo observatory</a> with the <a href="https://en.wikipedia.org/wiki/Arecibo_Telescope" rel="nofollow noreferrer">Arecibo dish antenna</a> firmly planted on the ground that steered in different directions via the movable secondary + antenna on cables. (cf. <a href="https://astronomy.stackexchange.com/q/28508/7982">Did Arecibo&#39;s secondary optics compensate aberrations when viewing farther away from vertical?</a>)</p>
<p><a href="https://i.sstatic.net/8AxidaTK.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/8AxidaTK.jpg" alt="enter image description here" /></a>(<a href="https://commons.wikimedia.org/wiki/File:Arecibo_radio_telescope_SJU_06_2019_6144.jpg" rel="nofollow noreferrer">source</a>)</p>
<p>and of course the relatively new and <em>essentially static</em> <a href="https://en.wikipedia.org/wiki/Five-hundred-meter_Aperture_Spherical_Telescope" rel="nofollow noreferrer">Five-hundred-meter Aperture Spherical Telescope</a> (cf. <a href="https://astronomy.stackexchange.com/q/16240/7982">What is a radio &quot;homology telescope&quot; and is the 500m dish in China one?</a>) which while it changes shape (and actually effective aperture) when it moves its primary focus, does not overcome <span class="math-container">$\cos \theta$</span> as it steers.</p>
<p><a href="https://i.sstatic.net/pBdgYQPf.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/pBdgYQPf.jpg" alt="Five-hundred-meter Aperture Spherical Telescope" /></a>(<a href="https://commons.wikimedia.org/wiki/File:FAST_Radio_Telescope_(captured_from_video).jpg" rel="nofollow noreferrer">source</a>)</p>
<p><strong>An example of an interferometric array with fixed, non steering elements</strong> would be the <a href="https://en.wikipedia.org/wiki/Low-Frequency_Array" rel="nofollow noreferrer">LOFAR array</a> (cf. <a href="https://astronomy.stackexchange.com/q/48658/7982">Why is it called &quot;The LOFAR &#39;superterp&#39;&quot;? What is a terp, and what&#39;s so super about this one?</a>) where, as the annotation in the image indicates, the field of view is so large that a simple <span class="math-container">$\cos \theta$</span> term isn't enough and the realistic radiation pattern of the fixed antenna must be taken into account.</p>
<p><a href="https://i.sstatic.net/pAtyb.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/pAtyb.jpg" alt="LOFAR array" /></a> (<a href="https://arxiv.org/abs/1104.1577" rel="nofollow noreferrer">source</a>)</p>

CC BY-SA 4.0
2024-12-25T01:57:50.103Z
2024-12-25T01:57:50.103Z
uhoh
<p>I am studying the notes provided by NRAO (Essential Radio Astronomy).</p>
<p><a href="https://www.cv.nrao.edu/%7Esransom/web/Ch3.html#S1.SS5" rel="nofollow noreferrer">https://www.cv.nrao.edu/~sransom/web/Ch3.html#S1.SS5</a></p>
<p>I do not understand how the two-element quasi-monochromatic interferometer works. Consider a point source observed by the two dishes on their axes, with an angle <span class="math-container">$\theta$</span> wrt the baseline <span class="math-container">$b$</span> connecting them. The geometric delay <span class="math-container">$\tau_g$</span> of the arrival of a plane wavefront to one antenna wrt the other is given by</p>
<p><span class="math-container">$\tau_g = \frac{b \cos\theta}{c}$</span>,</p>
<p>where <span class="math-container">$c$</span> is the speed of light. The output voltages (suppose the antennas are identical)</p>
<p><span class="math-container">$V_1 = V \cos(\omega t)$</span>,</p>
<p><span class="math-container">$V_2 = V \cos[\omega (t - \tau_g)]$</span></p>
<p>are multiplied and time averaged yielding the response</p>
<p><span class="math-container">$R = (V^2/2) \cos\phi$</span>,</p>
<p>where <span class="math-container">$\phi = \omega \tau_g = (2 \pi b \cos\theta)/\lambda$</span>.</p>
<p>Then, since the amplitude <span class="math-container">$V^2 \propto S_\nu (A_1 A_2)^{1/2}$</span>, where <span class="math-container">$S_\nu$</span> is the flux density of the source and <span class="math-container">$A_{1,2}$</span> are the effective areas of the antennas, and since for the effective area we have (consider a 1D uniformly illuminated aperture of size <span class="math-container">$D$</span> for simplicity)</p>
<p><span class="math-container">$A \propto \mathrm{sinc}^2(\frac{D}{\lambda} \cos\theta)$</span>,</p>
<p>we finally have for the response:</p>
<p><span class="math-container">$R \propto \mathrm{sinc}^2(\frac{D}{\lambda} \cos\theta) \times \cos(\frac{2 \pi b \cos\theta}{\lambda})$</span>.</p>
<p>Now, I do not understand why we have to consider the variation with <span class="math-container">$\theta$</span> of the effective area of the antennas, since we always observe the source on the axes of the dishes. <span class="math-container">$\theta$</span> changes because the Earth rotates and so the geometric delay changes, but the dishes move to hold the source stable on their axes, so we should always have  the same effective area, and not a variation with <span class="math-container">$\theta$</span>. Thank you in andvance for your answers!</p>

CC BY-SA 4.0
2024-12-25T23:08:05.840Z
I added the link!
Al01
CC BY-SA 4.0
2024-12-26T15:06:25.827Z
“with an angle θ wrt the baseline b connecting them“, not the angle wrt the dish axes.  It is the time delay that is relevant.
eshaya
CC BY-SA 4.0
2024-12-26T15:06:41.640Z
@eshaya this does also have $A_1, A_2$ as "the effective areas of the antennas" and now I wonder if the $A \propto $ is being confused with the area $A$s? In that case the premise of the question is undermined?
uhoh
CC BY-SA 4.0
2024-12-26T15:12:23.993Z
@Al01  "Now, I do not understand why we have to consider the variation with 
 of the effective area of the antennas" I've added an answer giving examples where you would have, but we don't think your linked tutorial actually mentions that anywhere. Can you point to the specific place where it does? *Thanks!*
uhoh
CC BY-SA 4.0
2024-12-27T15:51:20.897Z
Equations 3.173 to 3.176 indicate that $\theta$ just refers to baseline angle.  The diagram shows a steerable 2 dish interferometer.
eshaya
CC BY-SA 4.0
2024-12-27T16:22:28.220Z
@uhoh, I have taken the second part involving the effective area of the antennas from some notes by my professor (I do not share them here because are not publicly available). The formula is shown together with a picture showing steerable dishes two element interferometer (as in the link I provided).Essentially the line of reasoning is as I wrote, without specifying if that formula applies or not non-steerable dishes. That's why I am confused and decided to ask this question here
Al01
CC BY-SA 4.0
2024-12-27T16:25:27.420Z
My opinion is that this formula applies only to the non-steerable case in which apertures have axes which point vertically. In this case the angle between the vertical and the source is $z = 90- \theta$, so since the effective area of the aperture is $A \propto sinc^2(\frac{D}{\lambda} \sin(z))$ you recover the formula I wrote in the above question.
Al01
CC BY-SA 4.0
2024-12-28T03:03:38.503Z
@Al01 I'm sorry, this has gotten too confusing now that we're talking about a secret equation that can't be shown. I don't really know exactly what question you are asking or exactly what you want as an answer as it currently stands.
uhoh
<radio-astronomy><radio-telescope><interferometry><education>
Two-element quasi-monochromatic multiplying interferometer- how does it work?
Al01
# Two-element quasi-monochromatic multiplying interferometer- how does it work?

(Asked by: username_0 on 2024-12-23)

I am studying the notes provided by NRAO (Essential Radio Astronomy).

[https://www.cv.nrao.edu/~sransom/web/Ch3.html#S1.SS5](https://www.cv.nrao.edu/%7Esransom/web/Ch3.html#S1.SS5)

I do not understand how the two-element quasi-monochromatic interferometer works. Consider a point source observed by the two dishes on their axes, with an angle $\theta$ wrt the baseline $b$ connecting them. The geometric delay $\tau\_g$ of the arrival of a plane wavefront to one antenna wrt the other is given by

$\tau\_g = \frac{b \cos\theta}{c}$,

where $c$ is the speed of light. The output voltages (suppose the antennas are identical)

$V\_1 = V \cos(\omega t)$,

$V\_2 = V \cos[\omega (t - \tau\_g)]$

are multiplied and time averaged yielding the response

$R = (V^2/2) \cos\phi$,

where $\phi = \omega \tau\_g = (2 \pi b \cos\theta)/\lambda$.

Then, since the amplitude $V^2 \propto S\_\nu (A\_1 A\_2)^{1/2}$, where $S\_\nu$ is the flux density of the source and $A\_{1,2}$ are the effective areas of the antennas, and since for the effective area we have (consider a 1D uniformly illuminated aperture of size $D$ for simplicity)

$A \propto \mathrm{sinc}^2(\frac{D}{\lambda} \cos\theta)$,

we finally have for the response:

$R \propto \mathrm{sinc}^2(\frac{D}{\lambda} \cos\theta) \times \cos(\frac{2 \pi b \cos\theta}{\lambda})$.

Now, I do not understand why we have to consider the variation with $\theta$ of the effective area of the antennas, since we always observe the source on the axes of the dishes. $\theta$ changes because the Earth rotates and so the geometric delay changes, but the dishes move to hold the source stable on their axes, so we should always have the same effective area, and not a variation with $\theta$. Thank you in andvance for your answers!

**Tags:** radio-astronomy, radio-telescope, interferometry, education
> **Comments:**
> **username_0**: I added the link!  
> **username_1**: “with an angle θ wrt the baseline b connecting them“, not the angle wrt the dish axes.  It is the time delay that is relevant.  
> **username_2**: @eshaya this does also have $A_1, A_2$ as "the effective areas of the antennas" and now I wonder if the $A \propto $ is being confused with the area $A$s? In that case the premise of the question is undermined?  
> **username_2**: @Al01  "Now, I do not understand why we have to consider the variation with   of the effective area of the antennas" I've added an answer giving examples where you would have, but we don't think your linked tutorial actually mentions that anywhere. Can you point to the specific place where it does? *Thanks!*  
> **username_1**: Equations 3.173 to 3.176 indicate that $\theta$ just refers to baseline angle.  The diagram shows a steerable 2 dish interferometer.  
> **username_0**: @uhoh, I have taken the second part involving the effective area of the antennas from some notes by my professor (I do not share them here because are not publicly available). The formula is shown together with a picture showing steerable dishes two element interferometer (as in the link I provided).Essentially the line of reasoning is as I wrote, without specifying if that formula applies or not non-steerable dishes. That's why I am confused and decided to ask this question here  
> **username_0**: My opinion is that this formula applies only to the non-steerable case in which apertures have axes which point vertically. In this case the angle between the vertical and the source is $z = 90- \theta$, so since the effective area of the aperture is $A \propto sinc^2(\frac{D}{\lambda} \sin(z))$ you recover the formula I wrote in the above question.  
> **username_2**: @Al01 I'm sorry, this has gotten too confusing now that we're talking about a secret equation that can't be shown. I don't really know exactly what question you are asking or exactly what you want as an answer as it currently stands.  

---

## 1 answers

### Answer by username_2 on 2024-12-25. Score: 0

> ...since we always observe the source on the axes of the dishes...

*Optical axis*, often. Mechanical/structural axis, ***not so fast!***

Radio astronomy is full of large antennas that are physically static, but electronically steered, and there are some fixed dishes where only the secondary reflector moves.

All of those will present a $\cos \theta$ like cross-section to the source.

**For 100% static antennas** a recent example is [CHIME](https://en.wikipedia.org/wiki/Canadian_Hydrogen_Intensity_Mapping_Experiment) (cf. [How does Canada's radiotelescope CHIME manage polarization-dependent apodization of the fast f/0.25 optics?](https://astronomy.stackexchange.com/q/28615/7982))

[![CHIME](https://i.sstatic.net/1cVDT.jpg)](https://i.sstatic.net/1cVDT.jpg)[source](https://x.com/SKAO/status/1065277565429010432?mx=2)

and a famous, historic older and longer wavelength example is the [Interplanetary Scintillation Array](https://en.wikipedia.org/wiki/Interplanetary_Scintillation_Array) where the first pulsar was discovered by then graduate student [Jocelyn Bell Burnell](https://en.wikipedia.org/wiki/Jocelyn_Bell_Burnell) through careful and painstaking review of chart recorder data. (cf. [What exactly is interplanetary scintillation; what was the Interplanetary Scintillation Array looking for? Did it successfully observe any?](https://astronomy.stackexchange.com/q/34532/7982))

[![Interplanetary Scintillation Array](https://i.sstatic.net/TplAAMrJ.jpg)](https://i.sstatic.net/TplAAMrJ.jpg)([source](https://commons.wikimedia.org/wiki/File:Cmglee_Cambridge_MRAO_IPS_Array.jpg))

and the [Ohio State University Radio Observatory](https://en.wikipedia.org/wiki/Ohio_State_University_Radio_Observatory) of [Wow! signal](https://en.wikipedia.org/wiki/Wow!_signal) fame (cf. [What was the field of view of the Ohio State University Radio Observatory of Wow! signal fame?](https://astronomy.stackexchange.com/q/35091/7982) and [Did comets 266P/Christensen or P/2008 Y2 (Gibbs) cause the Wow! signal?](https://astronomy.stackexchange.com/q/35089/7982))

[![enter image description here](https://i.sstatic.net/OlkSVdI1.jpg)](https://i.sstatic.net/OlkSVdI1.jpg)([source](https://commons.wikimedia.org/wiki/File:%D0%A0%D0%B0%D0%B4%D0%B8%D0%BE%D1%82%D0%B5%D0%BB%D0%B5%D1%81%D0%BA%D0%BE%D0%BF_%D0%A3%D0%BD%D0%B8%D0%B2%D0%B5%D1%80%D1%81%D0%B8%D1%82%D0%B5%D1%82%D0%B0_%D0%9E%D0%B3%D0%B0%D0%B9%D0%BE.jpg))

For static dishes, look no further than the late great [Arecebo observatory](https://en.wikipedia.org/wiki/Arecibo_Observatory) with the [Arecibo dish antenna](https://en.wikipedia.org/wiki/Arecibo_Telescope) firmly planted on the ground that steered in different directions via the movable secondary + antenna on cables. (cf. [Did Arecibo's secondary optics compensate aberrations when viewing farther away from vertical?](https://astronomy.stackexchange.com/q/28508/7982))

[![enter image description here](https://i.sstatic.net/8AxidaTK.jpg)](https://i.sstatic.net/8AxidaTK.jpg)([source](https://commons.wikimedia.org/wiki/File:Arecibo_radio_telescope_SJU_06_2019_6144.jpg))

and of course the relatively new and *essentially static* [Five-hundred-meter Aperture Spherical Telescope](https://en.wikipedia.org/wiki/Five-hundred-meter_Aperture_Spherical_Telescope) (cf. [What is a radio "homology telescope" and is the 500m dish in China one?](https://astronomy.stackexchange.com/q/16240/7982)) which while it changes shape (and actually effective aperture) when it moves its primary focus, does not overcome $\cos \theta$ as it steers.

[![Five-hundred-meter Aperture Spherical Telescope](https://i.sstatic.net/pBdgYQPf.jpg)](https://i.sstatic.net/pBdgYQPf.jpg)([source](https://commons.wikimedia.org/wiki/File:FAST_Radio_Telescope_(captured_from_video).jpg))

**An example of an interferometric array with fixed, non steering elements** would be the [LOFAR array](https://en.wikipedia.org/wiki/Low-Frequency_Array) (cf. [Why is it called "The LOFAR 'superterp'"? What is a terp, and what's so super about this one?](https://astronomy.stackexchange.com/q/48658/7982)) where, as the annotation in the image indicates, the field of view is so large that a simple $\cos \theta$ term isn't enough and the realistic radiation pattern of the fixed antenna must be taken into account.

[![LOFAR array](https://i.sstatic.net/pAtyb.jpg)](https://i.sstatic.net/pAtyb.jpg) ([source](https://arxiv.org/abs/1104.1577))


---

