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If a star is close enough that we can measure its parallax and that it doesn't appear point-like, then we can deduce its diameter. Can we measure the diameter of all the stars for which we can measure the parallax, or is the angular diameter a limiting factor? How far out is the diameter of a star measurable?

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    $\begingroup$ When parallax fails, we can use spectral output analysis (both shape and power) to estimate the star's classification, which leads to at least a crude estimate of diameter. $\endgroup$ – Carl Witthoft Apr 3 at 13:38
  • $\begingroup$ There is probably a good way to estimate diameter for even distant Cepheid variables: en.m.wikipedia.org/wiki/Cepheid_variable $\endgroup$ – Wayfaring Stranger Apr 4 at 16:04
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It's neither the angular diameter or prallax precision that is the limiting factor, but the fact that it is difficult to get the interferometric measurements for faint stars.

State-of-the-art angular diameters are measured by infrared interferometry (e.g. with the CHARA array - Gordon et al. 2019). The most precise measurements of angular diameters have an uncertainty of about 17 microarcseconds in that paper. That means the smallest angular diameters that can yield a solid detection of the disc of a star are about $\theta = 50$ microarcseconds. But such measurements can only be achieved for stars down to about $V \sim 6$.

We can write down an equation for the relationship between $\theta$, the radius of the star $R$ and its distance $D$. $$ D = \frac{2R}{\theta} = 186\left(\frac{R}{R_{\odot}}\right) \left(\frac{\theta}{50\ \mu{\rm as}}\right)^{-1}\ {\rm pc}$$

This means that if a $1R_{\odot}$ star has its distance known more precisely than the $\sim 30$% error in the angular diameter, then it's radius can be measured out to distances of about 200 pc (with a precision of about 30%). This is easily achieved - a $1R_{\odot}$ solar-type star at a distance of 200 pc has a brightness of $V \sim 11$ and has a parallax uncertainty in the Gaia catalogue of only about 0.1 mas (mostly a systematic error at present), meaning the distance is known to about 2%. Thus, parallax uncertainty is not the limiting factor. However, a 6th magnitude solar-type star would be at $D \sim 20$ pc. Much further away than this and the source is just not bright enough to get an angular diameter measurement with CHARA.

On the other hand, a large red giant star, with $R \sim 200 R_{\odot}$ could in principle have its radius measured to 30% out to a distance of 37 kpc with the angular resolution available. Such a star would not however have a measurable parallax at present. The parallax would be 27 microarcseconds, which should just be resolvable in the final Gaia data reduction in a few years time. However, the star would also have $V \sim 14$ and is way too faint for the current interferometric capabilities of CHARA. A 6th magnitude red giant is at $D \sim 500$ pc, a distance at which both its angular diameter and parallax are precisely measurable.

So the answer is that the limitation is not the angular resolution, or the parallax. It is the brightness limit for the interferometers that are used to do the measurements. This limitation means that you can measure the diamters of solar-type dwarfs to a few 10s of pc, and the diameters of the largest red giants to about 1 kpc.

Update: There are other interferometers that can be used. The VLTI/Amber interferometer in Chile utilises larger telescopes than CHARA and can in principle work to fainter magnitudes. A paper by Chesneau et al. (2014) measures an angular diameter for HR5171A (a red hypergiant) to be $3.39 \pm 0.02$ mas, at a distance of 3.6 kpc. https://www.aanda.org/articles/aa/full_html/2014/03/aa22421-13/aa22421-13.html

Although much further away, this huge star is bright enough ($V \sim 6.5$) that an angular diameter measurement can still be achieved. Note though that the limitation here is still the brightness of the object, not its distance or angular size.

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The parallax is always easier to measure than the angular size of any planet. This is also true for most stars, excluding hypergiants and some supergiants. The parallax is given by

$ \displaystyle \theta_{p} = \frac{d_e}{d}$

where $d_e$ is the Earth-Sun distance and is $d$ the distance of the star. Instead, the angular size of an object with radius $r$ is

$ \displaystyle \theta_{r} = \frac{r}{d}$

Let $\theta_{min,p}$ and $\theta_{min, r}$ be the best angular resolutions we can obtain for a measurement of parallax and angular size respectively. Clearly, the angular size will be the limiting factor if

$\displaystyle \frac{r}{\theta_{min,r}} < \frac{d_e}{\theta_{min,p}}$

Now, let us take $\theta_{min,p} \approx 10^{-5}$ as (precision of Gaia mission) and $\theta_{min,r} \approx 2 \cdot 10^{-6}$ as (The Navy Prototype Optical Interferometer). Therefore, the condition for the angular size to be the limiting factor is (to the closest order of magnitude):

$r < d_e/10$

Of course there can be no planet with a radius larger than a tenth the Earth-Sun distance, therefore the angular size is always the limiting factor for planets.

The largest main sequence stars (O-type stars) have radii of about $20 R_{⊙}$, while the Earth-Sun distance is just over $200 R_{⊙}$. Therefore, O-type stars sit across the boundary, whereas the angular size is the limiting factor for all other main sequence stars.

Now, let's examine stars outside the main sequence. The largest stars are the Hypergiants, with radii greater than $1000 R_{⊙}$. The largest known star to date is VY Canis Majoris, with a radius of $1,420 R_{⊙}$. Supergiants usually range from $50 R_{⊙}$ to $500 R_{⊙}$, so in this case the distance is the limiting factor. Using the best interferometers to date, we would be able to measure diameters up to 200 $\mu$as with a precision of 1%. This corresponds to measuring the diameter of a typical supergiant up to a distance of 2kpc. The first star to have its angular diameter measured was Betelgeuse.

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  • $\begingroup$ The largest stars, for which radii can be determined a tthe largest distances, have radii larger than 1 au, so it is unclear how your argument works. Gaia measures parallaxes, not angular diameters, and does not have an angular resolution of 20 microarcseconds. An answer needs to talk about the capabilities of interferometry, which is how angula rdiameters are determined. $\endgroup$ – Rob Jeffries Apr 3 at 13:55
  • $\begingroup$ @RobJeffries could you link a source for the precision of Gaia? Fundamental Astronomy pag. 32 states that it is on the order of 10^(-5) arc seconds. $\endgroup$ – Flaffo Apr 3 at 15:44
  • $\begingroup$ The precision with which Gaia can find the position of a star is only loosely related to its angular resolution which is of order a few tenths of an arcsecond. Can you point to any paper where Gaia has measured the diameter of a star? $\endgroup$ – Rob Jeffries Apr 3 at 16:28
  • $\begingroup$ @RobJeffries I have edited my answer now, so the comparison only uses the angular resolution in the two cases. On the other hand, the The Navy Prototype Optical Interferometer claims it can perform measurements up to 200uas with precision of 1%. $\endgroup$ – Flaffo Apr 3 at 16:50

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