<p>Building on the answers from @anders-sandberg and @ProfRob I think I get an answer for my own question. I am most certainly reinventing the wheel, but it has been fun. Please comment or edit if you know any pointer on a similar derivation.</p>
<p>As a (very) rough approximation, one may assume that in a given line of sight, there is locally a uniform, independent and identically distributed density of stars. This can be modeled for instance by a <a href="https://en.wikipedia.org/wiki/Poisson_distribution" rel="nofollow noreferrer">3D Poisson distribution</a>. A 3D Poisson distribution consists in drawing random positions in space independently for the 3 axis and independently for the different points.</p>
<p>As a consequence, there are stars in all directions, but they have different luminances due to their distance. It is easy to derive the probability distribution of the distance for such a model and by a change of variable, one can find that of luminance as :</p>
<p><span class="math-container">$$
\text{Pr}(L) \propto \frac 1 {d^{\frac 5 2}}
$$</span></p>
<p>This is what you get with a simple simulation by placing stars at random positions in the universe but also if you plot the distribution of the luminance for stars drawn from <a href="https://github.com/astronexus/HYG-Database" rel="nofollow noreferrer">a catalog of more than 100k stars</a> (left panel):</p>
<p><a href="https://i.sstatic.net/GjQ06.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/GjQ06.jpg" alt="result" /></a></p>
<p>In addition, it is also a way to build a very simple model for generating images of the night sky (right panel).</p>
<p>The full details of my musings are available in this <a href="https://laurentperrinet.github.io/sciblog/posts/2021-03-27-density-of-stars-on-the-surface-of-the-sky.html" rel="nofollow noreferrer">blog post</a>.</p>
<p>PS: the other part of my question (can such a model be used to fit the density of stars in different areas of the sky?) will be for another time...</p>
<p>PS2: There is a relation with <a href="https://en.wikipedia.org/wiki/Olbers%27_paradox" rel="nofollow noreferrer">Olbers' paradox</a> which I discuss in the post...</p>

CC BY-SA 4.0
2021-05-05T09:07:58.240Z
2021-05-31T14:23:45.603Z
2021-05-31T14:23:45.603Z
meduz
<p>If stars are uniformly distributed with density <span class="math-container">$\rho$</span> per cubic parsec, out to a distance <span class="math-container">$d$</span> parsecs, then that volume contains <span class="math-container">$4\pi \rho d^3/3$</span> stars spread over the whole sky.</p>
<p>The angular size of the whole sky is 41,252.96125 square degrees, so you would divide by that number to estimate the stellar density in stars per square degree.</p>
<p>This assumes that you can see all the stars. In practice you cannot, because of dust extinction and because stars become fainter the further away they are. You can account for these in any calculation but the details depend on the luminosity function of the stars, the distribution of dust and the instrumental properties of the telescope/detector used to take the picture.</p>

CC BY-SA 4.0
2021-03-27T14:43:13.007Z
2021-03-27T14:43:13.007Z
ProfRob
<p>If we consider a 3D Poisson distribution over some 3D volume <span class="math-container">$V$</span>, the rate/density of the orthogonal projection onto a plane will at each point will be the 3D rare <span class="math-container">$\lambda$</span> multiplied by integrated length along the projection that is inside <span class="math-container">$V$</span>. So a unit cube will produce a square with the same rate if projected along one of its axes (and a hexagon with uneven rate along other directions). If the rate is spatially varying <span class="math-container">$\lambda(x,y,z)$</span> the projection will just be <span class="math-container">$\lambda(x,y)=\int \lambda(x,y,z) dz$</span>.</p>
<p>Note that an infinite space with finite <span class="math-container">$\lambda$</span> will produce an infinite point density. Real skies (1) have uneven rates so that most rays stop finding new stars at the edge of the galaxy and remote galaxies are sparse enough that they produce a finite projection, (2) the finite age of the universe allows setting an upper distance limit, and (3) you are actually interested in stars of a certain magnitude range, and remote stars will be dimmed by distance. That means that there is a distance fading factor <span class="math-container">$f(z)$</span> multiplying the rate, <span class="math-container">$\lambda(x,y)=\int \lambda(x,y,z) f(z) dz$</span>.</p>

CC BY-SA 4.0
2021-03-27T14:50:10.530Z
2021-03-27T14:50:10.530Z
Anders Sandberg
<p>The pattern of stars on the surface of the <a href="https://simple.wikipedia.org/wiki/Celestial_sphere" rel="nofollow noreferrer">celestial sphere</a> seems perceptually to have an universally looking structure. If you pick a region of the sky at random, for instance by using the (excellent) set of tools from the <a href="https://stellarium-web.org/" rel="nofollow noreferrer">stellarium suite</a> you get for example this:</p>
<p><a href="https://i.sstatic.net/EQwxT.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/EQwxT.jpg" alt="an area of the sky used for illustration purposes" /></a></p>
<p>There are bigger structures, like the milky way or more clustered areas, but on a smaller scale (like that you will see in binoculars with a field of view of the size of the moon), you will <em>very often</em> get this similar pattern. This looks to me very much like a self-similar fractal and I am curious to know if anybody knows more about it.</p>
<p>Assuming a given density of stars in this patch of the universe, and  omitting inhomogeneities at first,  <strong>can we derive a mathematical expression for describing the density of stars in a given patch of the sky</strong> ?</p>
<p>As a bonus, is there a map estimating this density for different areas of the <a href="https://simple.wikipedia.org/wiki/Celestial_sphere" rel="nofollow noreferrer">celestial sphere</a>? Clearly, the milky way is more crowded and would perhaps have more structure...</p>

CC BY-SA 4.0
2021-03-27T14:44:17.000Z
What is meant by a 3D Poisson distribution?
ProfRob
CC BY-SA 4.0
2021-03-27T15:18:41.313Z
Because the density is completely overshadowed by the galactic structures, the maths equation of views from within a globular/spiral are almost entirely described by the mass distribution of the galaxy relative to the viewpoint, i.e. the galaxy is the maths formula in our view, so why search for a inhomohegeneous one? I dont believe there is a formula that doesn't mostly stem from galactic geometry.
bandybabboon
CC BY-SA 4.0
2021-03-27T15:37:10.923Z
Isn't the mass equation a constant number relative to volume, excepting gradients due to the galaxy?
bandybabboon
CC BY-SA 4.0
2021-03-28T14:52:50.277Z
A *guest* mentioned [Olber's paradox](https://en.wikipedia.org/wiki/Olbers%27_paradox) and I think that term is indeed worth mentioning.
B--rian
CC BY-SA 4.0
2021-04-09T07:12:18.217Z
A 3D distribution consists in drawing random positions in space independently for the 3 axis and independently for the differents points. I will update my question to clarify this.
meduz
CC BY-SA 4.0
2021-04-09T07:14:00.673Z
Thanks a lot for your comments - I will try to clarify my question to make it clearer! (it's not an easy task to make a truely easy questions look easy!)
meduz
<universe><space><night-sky>
Density of stars on celestial sphere
meduz
# Density of stars on celestial sphere

(Asked by: username_0 on 2021-03-27)

The pattern of stars on the surface of the [celestial sphere](https://simple.wikipedia.org/wiki/Celestial_sphere) seems perceptually to have an universally looking structure. If you pick a region of the sky at random, for instance by using the (excellent) set of tools from the [stellarium suite](https://stellarium-web.org/) you get for example this:

[![an area of the sky used for illustration purposes](https://i.sstatic.net/EQwxT.jpg)](https://i.sstatic.net/EQwxT.jpg)

There are bigger structures, like the milky way or more clustered areas, but on a smaller scale (like that you will see in binoculars with a field of view of the size of the moon), you will *very often* get this similar pattern. This looks to me very much like a self-similar fractal and I am curious to know if anybody knows more about it.

Assuming a given density of stars in this patch of the universe, and omitting inhomogeneities at first, **can we derive a mathematical expression for describing the density of stars in a given patch of the sky** ?

As a bonus, is there a map estimating this density for different areas of the [celestial sphere](https://simple.wikipedia.org/wiki/Celestial_sphere)? Clearly, the milky way is more crowded and would perhaps have more structure...

**Tags:** universe, space, night-sky
> **Comments:**
> **username_1**: What is meant by a 3D Poisson distribution?  
> **username_2**: Because the density is completely overshadowed by the galactic structures, the maths equation of views from within a globular/spiral are almost entirely described by the mass distribution of the galaxy relative to the viewpoint, i.e. the galaxy is the maths formula in our view, so why search for a inhomohegeneous one? I dont believe there is a formula that doesn't mostly stem from galactic geometry.  
> **username_2**: Isn't the mass equation a constant number relative to volume, excepting gradients due to the galaxy?  
> **username_3**: A *guest* mentioned [Olber's paradox](https://en.wikipedia.org/wiki/Olbers%27_paradox) and I think that term is indeed worth mentioning.  
> **username_0**: A 3D distribution consists in drawing random positions in space independently for the 3 axis and independently for the differents points. I will update my question to clarify this.  
> **username_0**: Thanks a lot for your comments - I will try to clarify my question to make it clearer! (it's not an easy task to make a truely easy questions look easy!)  

---

## 3 answers

### Answer by username_0 on 2021-05-05. Score: 0 | Accepted answer

Building on the answers from @anders-sandberg and @ProfRob I think I get an answer for my own question. I am most certainly reinventing the wheel, but it has been fun. Please comment or edit if you know any pointer on a similar derivation.

As a (very) rough approximation, one may assume that in a given line of sight, there is locally a uniform, independent and identically distributed density of stars. This can be modeled for instance by a [3D Poisson distribution](https://en.wikipedia.org/wiki/Poisson_distribution). A 3D Poisson distribution consists in drawing random positions in space independently for the 3 axis and independently for the different points.

As a consequence, there are stars in all directions, but they have different luminances due to their distance. It is easy to derive the probability distribution of the distance for such a model and by a change of variable, one can find that of luminance as :

$$
\text{Pr}(L) \propto \frac 1 {d^{\frac 5 2}}
$$

This is what you get with a simple simulation by placing stars at random positions in the universe but also if you plot the distribution of the luminance for stars drawn from [a catalog of more than 100k stars](https://github.com/astronexus/HYG-Database) (left panel):

[![result](https://i.sstatic.net/GjQ06.jpg)](https://i.sstatic.net/GjQ06.jpg)

In addition, it is also a way to build a very simple model for generating images of the night sky (right panel).

The full details of my musings are available in this [blog post](https://laurentperrinet.github.io/sciblog/posts/2021-03-27-density-of-stars-on-the-surface-of-the-sky.html).

PS: the other part of my question (can such a model be used to fit the density of stars in different areas of the sky?) will be for another time...

PS2: There is a relation with [Olbers' paradox](https://en.wikipedia.org/wiki/Olbers%27_paradox) which I discuss in the post...


---

### Answer by username_1 on 2021-03-27. Score: 3

If stars are uniformly distributed with density $\rho$ per cubic parsec, out to a distance $d$ parsecs, then that volume contains $4\pi \rho d^3/3$ stars spread over the whole sky.

The angular size of the whole sky is 41,252.96125 square degrees, so you would divide by that number to estimate the stellar density in stars per square degree.

This assumes that you can see all the stars. In practice you cannot, because of dust extinction and because stars become fainter the further away they are. You can account for these in any calculation but the details depend on the luminosity function of the stars, the distribution of dust and the instrumental properties of the telescope/detector used to take the picture.


---

### Answer by username_4 on 2021-03-27. Score: 3

If we consider a 3D Poisson distribution over some 3D volume $V$, the rate/density of the orthogonal projection onto a plane will at each point will be the 3D rare $\lambda$ multiplied by integrated length along the projection that is inside $V$. So a unit cube will produce a square with the same rate if projected along one of its axes (and a hexagon with uneven rate along other directions). If the rate is spatially varying $\lambda(x,y,z)$ the projection will just be $\lambda(x,y)=\int \lambda(x,y,z) dz$.

Note that an infinite space with finite $\lambda$ will produce an infinite point density. Real skies (1) have uneven rates so that most rays stop finding new stars at the edge of the galaxy and remote galaxies are sparse enough that they produce a finite projection, (2) the finite age of the universe allows setting an upper distance limit, and (3) you are actually interested in stars of a certain magnitude range, and remote stars will be dimmed by distance. That means that there is a distance fading factor $f(z)$ multiplying the rate, $\lambda(x,y)=\int \lambda(x,y,z) f(z) dz$.


---

