<p>Assuming that each (square) pixel has the same angular scale (not a given if the field of view is large) of $\theta$ degrees/pixel.</p>

<p>Then the declination (in degrees): $\delta \simeq \delta_0 + \theta y$ </p>

<p>The right ascension (in degrees): $\alpha \simeq \alpha_0 - \theta x/\cos \delta$</p>

<p>where $\alpha_0$ and $\delta_0$ are the RA and Dec at $x=0$, $y=0$. [The minus sign is there because right ascension increases towards the left of a sky image.]</p>

<p>This approximatimation becomes poor as the field of view gets larger.</p>

<p>A slightly more complex, but accurate, approach is described in <a href="http://gtn.sonoma.edu/data_reduction/astrometry.php" rel="nofollow">http://gtn.sonoma.edu/data_reduction/astrometry.php</a></p>

CC BY-SA 3.0
2016-03-15T22:43:25.230Z
Thanks Rob, this is the same formula I arrived at, but I can not find any source for this. Do you know where these steps might be mathematically demonstrated? A book, an article, anything that could be used as a source.
Gabriel
CC BY-SA 3.0
2016-03-15T23:06:49.407Z
A source? It's just from the definition of a spherical coordinate system. If a ship travels at constant speed this is how you would work out the latitude and longitude.
ProfRob
CC BY-SA 3.0
2016-03-15T23:17:37.317Z
No, [this](https://en.wikipedia.org/wiki/Spherical_coordinate_system#Definition) is the definition of a spherical coordinate system. I'm asking if you know of a source where this recipe is shown, as in *Section X: How to transform pixel coordinates to equatorial coordinates*. I'm still marking your answer as accepted, I'm just curious because I could not find an explicit presentation of such a basic set of equations for such a used procedure in Astronomy, in any book/article/site.
Gabriel
CC BY-SA 3.0
2016-03-16T02:01:51.927Z
Thank you for the source Rob! I'll take a look at it right now.
Gabriel
CC BY-SA 3.0
2016-03-15T22:16:58.143Z
2016-03-15T23:50:34.233Z
2016-03-15T23:50:34.233Z
ProfRob
<p>Say I have a FITS file with an observed field of stars, each one with x,y coordinates given in pixels. Assume the image is aligned so that the x axis is the right ascension, and the y axis the declination.</p>

<p>I'm looking for a tutorial to transform the pixel coordinates of each star to equatorial, even if the transformation is approximate.</p>

<p>I'v been looking for a while but found no tutorial/book/site that enumerates the steps needed to produce this transformation. </p>

<astrometry>
Transform pixel coordinates (in FITS file) to equatorial
Gabriel
# Transform pixel coordinates (in FITS file) to equatorial

(Asked by: username_0 on 2016-03-15)

Say I have a FITS file with an observed field of stars, each one with x,y coordinates given in pixels. Assume the image is aligned so that the x axis is the right ascension, and the y axis the declination.

I'm looking for a tutorial to transform the pixel coordinates of each star to equatorial, even if the transformation is approximate.

I'v been looking for a while but found no tutorial/book/site that enumerates the steps needed to produce this transformation.

**Tags:** astrometry

---

## 1 answers

### Answer by username_1 on 2016-03-15. Score: 3 | Accepted answer

Assuming that each (square) pixel has the same angular scale (not a given if the field of view is large) of $\theta$ degrees/pixel.

Then the declination (in degrees): $\delta \simeq \delta\_0 + \theta y$

The right ascension (in degrees): $\alpha \simeq \alpha\_0 - \theta x/\cos \delta$

where $\alpha\_0$ and $\delta\_0$ are the RA and Dec at $x=0$, $y=0$. [The minus sign is there because right ascension increases towards the left of a sky image.]

This approximatimation becomes poor as the field of view gets larger.

A slightly more complex, but accurate, approach is described in <http://gtn.sonoma.edu/data_reduction/astrometry.php>

> **Comments:**
> **username_0**: Thanks Rob, this is the same formula I arrived at, but I can not find any source for this. Do you know where these steps might be mathematically demonstrated? A book, an article, anything that could be used as a source.  
> **username_1**: A source? It's just from the definition of a spherical coordinate system. If a ship travels at constant speed this is how you would work out the latitude and longitude.  
> **username_0**: No, [this](https://en.wikipedia.org/wiki/Spherical_coordinate_system#Definition) is the definition of a spherical coordinate system. I'm asking if you know of a source where this recipe is shown, as in *Section X: How to transform pixel coordinates to equatorial coordinates*. I'm still marking your answer as accepted, I'm just curious because I could not find an explicit presentation of such a basic set of equations for such a used procedure in Astronomy, in any book/article/site.  
> **username_0**: Thank you for the source Rob! I'll take a look at it right now.  

---

