<p>Let's pretend for the moment that this is possible (it's generally not, as I'll explain below) and see how we would go about it.</p>

<p>In principle, this is basic trigonometry: you have a measured angle (<span class="math-container">$\alpha$</span>, the diameter of the star), you know the distance to the star (<span class="math-container">$D$</span>, from the six-month-separated parallax measurements), so to determine the linear diameter of the star (<span class="math-container">$d$</span>, in the same units as the distance) it's a simple matter of <span class="math-container">$\sin \alpha = d/D$</span>. In practice, the angles are small enough that you can use the small-angle approximation, so it's really just <span class="math-container">$\alpha \approx d/D \rightarrow d = \alpha D$</span> (assuming <span class="math-container">$\alpha$</span> is in radians).</p>

<p>As your book notes, you need to know the distance to the star; for nearby stars, this can be obtained by <a href="https://en.wikipedia.org/wiki/Stellar_parallax" rel="noreferrer">stellar parallax</a>, which involves the same basic trigonometric argument (only this time you know the "diameter", which is the size of the Earth's orbit).</p>

<p>So why is this, as I said, not generally possible? In practice, <a href="https://en.wikipedia.org/wiki/Angular_resolution" rel="noreferrer">the diameter of the <em>telescope</em> sets a lower limit on <em>observed</em> angular sizes</a> -- anything with a true diameter smaller than the telescope's angular limit will appear to have an angular diameter <span class="math-container">$\approx$</span> that limit. In addition, any telescope underneath the Earth's atmosphere will suffer the effects of blurring due to atmospheric turbulence, which sets an angular diameter limit of <span class="math-container">$\sim 0.5$</span> arc seconds (full-width half-maximum) or worse at the best sites. Since the <em>largest</em> stars in angular terms have <a href="https://www.eso.org/public/news/eso9706/" rel="noreferrer">diameters of about 0.05 arc seconds</a>, the result is that all stars in the image (photographic plate, CCD camera, etc.) will have the <em>same</em> observed angular diameter (e.g., 0.5 arc seconds at a really good site), regardless of their true diameters.</p>

<p>It <em>is</em> possible to do better for bright, nearby stars using large telescopes <em>in space</em> or special interferometric techniques -- see the second link in the previous paragraph for an example -- but this is going beyond the "stars on photographic plates" scenario your book seems to envisage.</p>

<p><em>And if for instance, using a 100'' telescope, you measure that a star has an apparent diameter of 0.66mm, can you directly compute the true diameter of the star?</em></p>

<p>Leaving aside the problems mentioned above, you don't have enough information. You need to convert the linear diameter in mm to an <em>angular</em> diameter, which means you need to know the plate scale (e.g., arcsec/mm) of the particular telescope + camera setup. You also need to know the distance to the star.</p>

CC BY-SA 4.0
2019-02-26T13:10:58.203Z
Ahhh the angular diameter makes much more sense! Thank you for the explanation. So for instance, in this paper https://apps.dtic.mil/dtic/tr/fulltext/u2/a067077.pdf, they measure the diameter of $\alpha$ Boo (I think this is Arcturus), and using your formula, I obain a "true" diameter of $\sim 4.37 \times 10^{10}$ vs $\sim 3.53 \times 10^{10}$ from Wikipedia. Does it look correct to you? Thank you again for the great answer!
Laurent Hayez
CC BY-SA 4.0
2019-02-26T13:32:22.310Z
P.S.: The book I'm reading explains the technique introduced by Labeyrie in 1970 ( http://adsabs.harvard.edu/full/1970A%26A.....6...85L ) to go around the problem that the blurring due to atmospheric effects is greater that the apparent diameter. I think he obtains the linear diameter that way, and from the plate scale he can deduce angular diameter. The book states that more than 30 stars had their diameters measured in this way :-).
Laurent Hayez
CC BY-SA 4.0
2019-02-26T13:35:58.850Z
@LaurentHayez When you say something has a value of such-and-such you need to state the *units*.  Without units it has no meaning, it's just a number.
StephenG - Help Ukraine
CC BY-SA 4.0
2019-02-26T13:40:14.130Z
@StephenG You're right. A bad habit I have from mathematics :-). I used meters in the first comment.
Laurent Hayez
CC BY-SA 4.0
2019-02-26T16:26:57.857Z
@LaurentHayez -- That looks reasonable, yes. (By the way, if you want a cleaner, typeset version of that paper, [here it is on the ADS](http://adsabs.harvard.edu/abs/1974ApJ...187..131C).)
Peter Erwin
CC BY-SA 4.0
2019-02-26T16:28:03.617Z
For what it's worth, [this 2016 paper by Chelli et al.](https://www.aanda.org/articles/aa/full_html/2016/05/aa27484-15/aa27484-15.html) mentions compiling a database of a little over 400 stars with accurate diameter measurements, most from interferometry and some from lunar occultation (measuring how rapidly the light from a star dims as the Moon passes in front of it -- a larger true angular diameter means a longer time to dim).
Peter Erwin
CC BY-SA 4.0
2019-02-26T20:09:14.550Z
For one nearby star (our sun) this works, as well as for moons and planets in our sun system.
Paŭlo Ebermann
CC BY-SA 4.0
2019-02-26T12:03:55.703Z
2019-02-26T12:03:55.703Z
Peter Erwin
<p>I am reading the book "Fourier Analysis" from T. W. Körner, and in chapter 95, he explains how you can compute the diameter of stars on a photographic plate. He says that "Since observations of the nearest stars at six-monthly intervals (i.e. using a diameter of the earth's orbit as a surveyor's base line) enable astronomers to measure the distance of these stars, knowledge of the apparent diameter (i.e. the diameters of the discs on the photographic plate) would then enable us to calculate the true diameters of the nearest stars". </p>

<p>So how do you calculate the true diameter of a star given its apparent diameter on a photographic plate? Is it related to the "zooming capacity" of the telescope one uses?</p>

<p>And if for instance, using a 100'' telescope, you measure that a star has an apparent diameter of 0.66mm, can you directly compute the true diameter of the star?</p>

<observational-astronomy><distances><photography>
Calculate the true diameter of stars from photographic plate
Laurent Hayez
# Calculate the true diameter of stars from photographic plate

(Asked by: username_0 on 2019-02-26)

I am reading the book "Fourier Analysis" from T. W. Körner, and in chapter 95, he explains how you can compute the diameter of stars on a photographic plate. He says that "Since observations of the nearest stars at six-monthly intervals (i.e. using a diameter of the earth's orbit as a surveyor's base line) enable astronomers to measure the distance of these stars, knowledge of the apparent diameter (i.e. the diameters of the discs on the photographic plate) would then enable us to calculate the true diameters of the nearest stars".

So how do you calculate the true diameter of a star given its apparent diameter on a photographic plate? Is it related to the "zooming capacity" of the telescope one uses?

And if for instance, using a 100'' telescope, you measure that a star has an apparent diameter of 0.66mm, can you directly compute the true diameter of the star?

**Tags:** observational-astronomy, distances, photography

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## 1 answers

### Answer by username_1 on 2019-02-26. Score: 12 | Accepted answer

Let's pretend for the moment that this is possible (it's generally not, as I'll explain below) and see how we would go about it.

In principle, this is basic trigonometry: you have a measured angle ($\alpha$, the diameter of the star), you know the distance to the star ($D$, from the six-month-separated parallax measurements), so to determine the linear diameter of the star ($d$, in the same units as the distance) it's a simple matter of $\sin \alpha = d/D$. In practice, the angles are small enough that you can use the small-angle approximation, so it's really just $\alpha \approx d/D \rightarrow d = \alpha D$ (assuming $\alpha$ is in radians).

As your book notes, you need to know the distance to the star; for nearby stars, this can be obtained by [stellar parallax](https://en.wikipedia.org/wiki/Stellar_parallax), which involves the same basic trigonometric argument (only this time you know the "diameter", which is the size of the Earth's orbit).

So why is this, as I said, not generally possible? In practice, [the diameter of the *telescope* sets a lower limit on *observed* angular sizes](https://en.wikipedia.org/wiki/Angular_resolution) -- anything with a true diameter smaller than the telescope's angular limit will appear to have an angular diameter $\approx$ that limit. In addition, any telescope underneath the Earth's atmosphere will suffer the effects of blurring due to atmospheric turbulence, which sets an angular diameter limit of $\sim 0.5$ arc seconds (full-width half-maximum) or worse at the best sites. Since the *largest* stars in angular terms have [diameters of about 0.05 arc seconds](https://www.eso.org/public/news/eso9706/), the result is that all stars in the image (photographic plate, CCD camera, etc.) will have the *same* observed angular diameter (e.g., 0.5 arc seconds at a really good site), regardless of their true diameters.

It *is* possible to do better for bright, nearby stars using large telescopes *in space* or special interferometric techniques -- see the second link in the previous paragraph for an example -- but this is going beyond the "stars on photographic plates" scenario your book seems to envisage.

*And if for instance, using a 100'' telescope, you measure that a star has an apparent diameter of 0.66mm, can you directly compute the true diameter of the star?*

Leaving aside the problems mentioned above, you don't have enough information. You need to convert the linear diameter in mm to an *angular* diameter, which means you need to know the plate scale (e.g., arcsec/mm) of the particular telescope + camera setup. You also need to know the distance to the star.

> **Comments:**
> **username_0**: Ahhh the angular diameter makes much more sense! Thank you for the explanation. So for instance, in this paper https://apps.dtic.mil/dtic/tr/fulltext/u2/a067077.pdf, they measure the diameter of $\alpha$ Boo (I think this is Arcturus), and using your formula, I obain a "true" diameter of $\sim 4.37 \times 10^{10}$ vs $\sim 3.53 \times 10^{10}$ from Wikipedia. Does it look correct to you? Thank you again for the great answer!  
> **username_0**: P.S.: The book I'm reading explains the technique introduced by Labeyrie in 1970 ( http://adsabs.harvard.edu/full/1970A%26A.....6...85L ) to go around the problem that the blurring due to atmospheric effects is greater that the apparent diameter. I think he obtains the linear diameter that way, and from the plate scale he can deduce angular diameter. The book states that more than 30 stars had their diameters measured in this way :-).  
> **username_2**: @LaurentHayez When you say something has a value of such-and-such you need to state the *units*.  Without units it has no meaning, it's just a number.  
> **username_0**: @StephenG You're right. A bad habit I have from mathematics :-). I used meters in the first comment.  
> **username_1**: @LaurentHayez -- That looks reasonable, yes. (By the way, if you want a cleaner, typeset version of that paper, [here it is on the ADS](http://adsabs.harvard.edu/abs/1974ApJ...187..131C).)  
> **username_1**: For what it's worth, [this 2016 paper by Chelli et al.](https://www.aanda.org/articles/aa/full_html/2016/05/aa27484-15/aa27484-15.html) mentions compiling a database of a little over 400 stars with accurate diameter measurements, most from interferometry and some from lunar occultation (measuring how rapidly the light from a star dims as the Moon passes in front of it -- a larger true angular diameter means a longer time to dim).  
> **username_3**: For one nearby star (our sun) this works, as well as for moons and planets in our sun system.  

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