<p>All images of stars (bar one or two of the closest or largest) are effectively those of <strong>point sources</strong>. The image we see is the convolution of a point with the instumental &quot;<a href="https://en.wikipedia.org/wiki/Point_spread_function" rel="noreferrer">point spread function</a>&quot;, which is, specifically, what a point-like source would look like if observed with that telescope.</p>
<p>The neutron star you discuss is not imaged by any telescope, in the sense of being able to resolve its disk. The image you see <em>is</em> the point spread function of the telescope. The derived angular diameter simply comes from an estimate of its radius and its distance.</p>

CC BY-SA 4.0
2024-01-26T11:54:39.047Z
2024-01-26T11:54:39.047Z
ProfRob
<p>With very few exceptions of the biggest and most nearby sources (like Betelgeuze with &lt;~50mas) we cannot resolve the angular diameter of stellar-type sources by direct imaging means. Precision of angular measurements and thus position of sources can still be quite a big higher as one can acquire an image and determine the center of the circular image of the diffraction disc of the object.</p>
<p>In the paper they do just that: they measure the distances via parallax to the neutron star and then use models of the star in order to derive further parameters. They determine the radius by assuming a blackbody radiator in the measured distance with the observed flux and thus conclude (emphasis mine):</p>
<blockquote>
<p>Obtaining the parallax of this neutron star is only a means to
an end. <strong>With a distance known to 7%, the largest uncertainty in
the radius of this neutron star lies in the atmospheric or surface
emission models</strong>. In the context of simple <strong>two blackbody models</strong>
the radius R∞ is 16.8+1.1
−1.4 km, where this uncertainty reflects
the statistical uncertainty in the distance (as discussed by Pons
et al. (2001), the radius depends critically on the ill-defined
temperature of the cool component). The true uncertainty is
dominated by systematics in the models. This is still consistent
with many equations of state, but combined with information
for X-ray bursts and quiescent low-mass X-ray binaries, it can
be used to significantly constrain the nuclear equation of state
(Steiner et al. 2010).</p>
</blockquote>
<p>Generally any image, even of a perfect point source with no angular extent, will appear as a circle, the so-called <a href="https://en.wikipedia.org/wiki/Airy_disk" rel="noreferrer">Airy-disc</a> due to diffraction on the opening of the optics itself. Nano-arcseconds is out of question even with positional determination for todays telescopes. The currently best positional accuracy is around what they show here or what e.g. Gaia produces with stellar location accuracies and parallaxes in the order of 10...20 µas; the best spatially-resolved imaging is of the order of 1...10 mas, thus 100...1000 times less accurate.</p>

CC BY-SA 4.0
2024-01-27T07:33:46.153Z
Good answer, and now I see although nano arcseconds are got from the theoretical calculations, this doesn't mean we are actually seeing in such tiny angular size. But instead we are observing these images from models that measure the Airy disc from the point source, which comes after resolving an image fitting from diffraction as seen from a telescope at Earth. So that then, the new angular diameter size is in the telescope, not in the sky. Am I right or would you edit out or add something else to my new self explanation?
omivela17
CC BY-SA 4.0
2024-01-26T09:22:34.330Z
2024-01-26T09:22:34.330Z
planetmaker
<p>The angular diameter doesn't come from imaging, but from photometry and spectroscopy. The observers estimated the surface brightness from the spectrum, and then computed how big the angular diameter needs to be to produce the observed flux.</p>

CC BY-SA 4.0
2024-01-26T02:56:43.077Z
2024-01-26T02:56:43.077Z
John Doty
<p>I have a big doubt about our allegedly nearest (X Ray isolated) neutron star, also known as the Walter star, one of the members of the &quot;Magnificent Seven stars&quot;: RX J185635-3754.</p>
<p>So I came across <a href="https://iopscience.iop.org/article/10.1088/0004-637X/724/1/669/pdf" rel="nofollow noreferrer">an article</a> that says that:</p>
<ol>
<li>its distance from Earth is <span class="math-container">$D=123^{+11}_{-15}$</span> pc (Walter <em>et al.,</em> 2010),</li>
<li>its measured parallax is <span class="math-container">$\pi = 8.16^{+0.9}_{\;-0.8}$</span> mas (<em>ibidem</em>),</li>
<li>its redshifted radius at infinity, measured by us is <span class="math-container">$R_\infty = 16.8^{+1.1}_{-1.4}$</span> km (<em>ibidem</em>)</li>
</ol>
<p>However, this would mean that an angular diameter of this object as measured by us is:</p>
<p><span class="math-container">$$2R_\infty/D \sim1.83 \text{   nas (yes, that's right, nano arc-seconds.)}$$</span></p>
<p>I googled with no much effort about this object and in Wikipedia there is a <a href="https://en.wikipedia.org/wiki/RX_J1856.5%E2%88%923754#/media/File:RX_J1856.5-3754.jpg" rel="nofollow noreferrer">beautiful X-ray image from the star</a>, and you can see (also inside the Wikipedia article after clicking on the attached link to) <a href="https://simbad.cds.unistra.fr/simbad/sim-id?Ident=1ES%201853-37.9" rel="nofollow noreferrer">the SIMBAD object data query too</a>, where you will see a tiny box where you can query the image of the object from XMM-Newton around a 1 arcmin large red spot and see a circle almost circling it. In both of these images it is possible to look at something, for sure. Or at least an excess I guess, which I don't know but I think maybe is due to saturation of light coming from the star (sorry, I'm afraid I know very little about observational astronomy and queries on SIMBAD).</p>
<p>My question is enlisted as one, in the next two (hope you all can get me):</p>
<p>I remember the angular diameters from the SMBHs in M87* and Sgr A* are in micro arcseconds, so why are we still getting an image from this star if its own is in nano arcseconds? What we are watching in both pictures is that excess I mentioned above? I mean, instead of watching the actual picture of the star from Earth? For example, for the red spot in the image query box in SIMBAD, if we wanted to see the actual star we would have to zoom more?</p>
<p>Or maybe my question and self explanation are silly and it is just all about the difference between parallax and angular diameter, or getting to learn how to use SIMBAD, but I would like to get an easy explanation about all this.</p>

<neutron-star><size><interferometry><angular-diameter>
About angular diameter, parallax and image of the nearest neutron star RX J185635-3754
omivela17
# About angular diameter, parallax and image of the nearest neutron star RX J185635-3754

(Asked by: username_0 on 2024-01-26)

I have a big doubt about our allegedly nearest (X Ray isolated) neutron star, also known as the Walter star, one of the members of the "Magnificent Seven stars": RX J185635-3754.

So I came across [an article](https://iopscience.iop.org/article/10.1088/0004-637X/724/1/669/pdf) that says that:

1. its distance from Earth is $D=123^{+11}\_{-15}$ pc (Walter *et al.,* 2010),
2. its measured parallax is $\pi = 8.16^{+0.9}\_{\;-0.8}$ mas (*ibidem*),
3. its redshifted radius at infinity, measured by us is $R\_\infty = 16.8^{+1.1}\_{-1.4}$ km (*ibidem*)

However, this would mean that an angular diameter of this object as measured by us is:

$$2R\_\infty/D \sim1.83 \text{ nas (yes, that's right, nano arc-seconds.)}$$

I googled with no much effort about this object and in Wikipedia there is a [beautiful X-ray image from the star](https://en.wikipedia.org/wiki/RX_J1856.5%E2%88%923754#/media/File:RX_J1856.5-3754.jpg), and you can see (also inside the Wikipedia article after clicking on the attached link to) [the SIMBAD object data query too](https://simbad.cds.unistra.fr/simbad/sim-id?Ident=1ES%201853-37.9), where you will see a tiny box where you can query the image of the object from XMM-Newton around a 1 arcmin large red spot and see a circle almost circling it. In both of these images it is possible to look at something, for sure. Or at least an excess I guess, which I don't know but I think maybe is due to saturation of light coming from the star (sorry, I'm afraid I know very little about observational astronomy and queries on SIMBAD).

My question is enlisted as one, in the next two (hope you all can get me):

I remember the angular diameters from the SMBHs in M87\* and Sgr A\* are in micro arcseconds, so why are we still getting an image from this star if its own is in nano arcseconds? What we are watching in both pictures is that excess I mentioned above? I mean, instead of watching the actual picture of the star from Earth? For example, for the red spot in the image query box in SIMBAD, if we wanted to see the actual star we would have to zoom more?

Or maybe my question and self explanation are silly and it is just all about the difference between parallax and angular diameter, or getting to learn how to use SIMBAD, but I would like to get an easy explanation about all this.

**Tags:** neutron-star, size, interferometry, angular-diameter

---

## 3 answers

### Answer by username_2 on 2024-01-26. Score: 8

With very few exceptions of the biggest and most nearby sources (like Betelgeuze with <~50mas) we cannot resolve the angular diameter of stellar-type sources by direct imaging means. Precision of angular measurements and thus position of sources can still be quite a big higher as one can acquire an image and determine the center of the circular image of the diffraction disc of the object.

In the paper they do just that: they measure the distances via parallax to the neutron star and then use models of the star in order to derive further parameters. They determine the radius by assuming a blackbody radiator in the measured distance with the observed flux and thus conclude (emphasis mine):

> Obtaining the parallax of this neutron star is only a means to
> an end. **With a distance known to 7%, the largest uncertainty in
> the radius of this neutron star lies in the atmospheric or surface
> emission models**. In the context of simple **two blackbody models**
> the radius R∞ is 16.8+1.1
> −1.4 km, where this uncertainty reflects
> the statistical uncertainty in the distance (as discussed by Pons
> et al. (2001), the radius depends critically on the ill-defined
> temperature of the cool component). The true uncertainty is
> dominated by systematics in the models. This is still consistent
> with many equations of state, but combined with information
> for X-ray bursts and quiescent low-mass X-ray binaries, it can
> be used to significantly constrain the nuclear equation of state
> (Steiner et al. 2010).

Generally any image, even of a perfect point source with no angular extent, will appear as a circle, the so-called [Airy-disc](https://en.wikipedia.org/wiki/Airy_disk) due to diffraction on the opening of the optics itself. Nano-arcseconds is out of question even with positional determination for todays telescopes. The currently best positional accuracy is around what they show here or what e.g. Gaia produces with stellar location accuracies and parallaxes in the order of 10...20 µas; the best spatially-resolved imaging is of the order of 1...10 mas, thus 100...1000 times less accurate.

> **Comments:**
> **username_0**: Good answer, and now I see although nano arcseconds are got from the theoretical calculations, this doesn't mean we are actually seeing in such tiny angular size. But instead we are observing these images from models that measure the Airy disc from the point source, which comes after resolving an image fitting from diffraction as seen from a telescope at Earth. So that then, the new angular diameter size is in the telescope, not in the sky. Am I right or would you edit out or add something else to my new self explanation?  

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### Answer by username_1 on 2024-01-26. Score: 7

All images of stars (bar one or two of the closest or largest) are effectively those of **point sources**. The image we see is the convolution of a point with the instumental "[point spread function](https://en.wikipedia.org/wiki/Point_spread_function)", which is, specifically, what a point-like source would look like if observed with that telescope.

The neutron star you discuss is not imaged by any telescope, in the sense of being able to resolve its disk. The image you see *is* the point spread function of the telescope. The derived angular diameter simply comes from an estimate of its radius and its distance.


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### Answer by username_3 on 2024-01-26. Score: 4

The angular diameter doesn't come from imaging, but from photometry and spectroscopy. The observers estimated the surface brightness from the spectrum, and then computed how big the angular diameter needs to be to produce the observed flux.


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