<p><strong>Stars</strong></p>
<p>One way of doing this is to take an estimate of the total number of stars in the observable universe, divide that by the volume of the observable universe to get a density, then multiply by some average cross-sectional area for a star.  The reciprocal of this would be a typical path length for a random sight line to intercept the surface of a star. Note that this is a static calculation which assumes the &quot;beam&quot; of radiation is a pulse short enough that stellar motions do not increase their effective cross-sectional area.</p>
<p>e.g., Using the estimates presented <a href="https://en.wikipedia.org/wiki/Observable_universe" rel="noreferrer">here</a>, the number of stars in the (observable) universe is <span class="math-container">$N\sim 10^{24}$</span>, the radius of the observable universe is <span class="math-container">$\sim 46.5$</span> billion light years, corresponding to a current volume of <span class="math-container">$V = 3.6 \times 10^{80}$</span> m<span class="math-container">$^3$</span>.</p>
<p>The next part is tricky. The average star is a low mass M dwarf with a radius of perhaps 30% that of the Sun. But a few percent of stars are giants and much larger - maybe 100 times the radius of the Sun and I think these will dominate in terms of average cross-sectional area. I.e.
<span class="math-container">$$\bar{\sigma} \sim \left( \frac{97 \times 0.3^2 + 3\times 100^2}{100}\right) \pi R_\odot^2 = 5\times 10^{20}\ {\rm m}^2\ .$$</span></p>
<p>Putting these together, the mean free path of a photon beam through the universe, before intercepting the surface of a star is
<span class="math-container">$$ l = \frac{V}{N\bar{\sigma}}\sim 10^{20}\ {\rm light\ years}$$</span>
and the chance of a random beam of light travelling the radius of the observable universe intercepting with a star is <span class="math-container">$\sim 46.5\times 10^9/10^{20}\sim 5\times10^{-9}$</span>.</p>
<p>This can only be an order of magnitude estimate. A better calculation could make a better demographic survey of stellar sizes (as a function of cosmic epoch), would take the expansion of the universe into account (in the future, the average stellar density would be lower) and that stars are not uniformly distributed,  but clumped into galaxies. It also ignores the Sun, which contributes an interception probability of <span class="math-container">$5\times 10^{-6}$</span> to a beam launched from Earth.</p>
<p><strong>Planets</strong></p>
<p>Even if we allow each star to have 10 planets, each planet is say 10-1000 times smaller than its star so contributes a fraction of the cross-sectional area. So the chances of light intercepting a planet are orders of magnitude lower than for stars.</p>
<p>(Note this ignores the trivial fact that a randomly directed beam launched from Earth has a 50% chance of hitting the Earth and a <span class="math-container">$5\times 10^{-6}$</span> chance of hitting the Moon.)</p>
<p><strong>Galaxies</strong></p>
<p>If we say our <span class="math-container">$10^{24}$</span> stars are actually in about <span class="math-container">$10^{14}$</span> galaxies, each with a radius of about 30 thousand light years (a bit smaller than the Milky Way), a similar calculation gives a mean free path of about a billion light years.</p>
<p><strong>Thus the overall conclusion is that a random light beam is likely to pass through many galaxies, but highly unlikely ever to intercept a star.</strong></p>

CC BY-SA 4.0
2024-12-22T13:12:48.470Z
Your laser beam is a mathematical line, right? With no cross-sectional area or divergence (or extinction). What are the probabilities like if we use a more realistic beam?
PM 2Ring
CC BY-SA 4.0
2024-12-22T18:28:20.647Z
Shouldn't the duration of beaming be taken into account? For instance if aiming a random spot on cmb, stars that have been missed could cross the continuous line later. Also if beam originates say from equator in a direction aligned with equatorial plane, there's 100% chances the beam crosses the Earth in the next ~12h
user721108
CC BY-SA 4.0
2024-12-22T18:32:37.790Z
@user721108 Yes, I'm sure that's right. You could make the answer much more complex. My calculation assumes an outwardly directed pulse.
ProfRob
CC BY-SA 4.0
2024-12-22T23:08:41.017Z
@PM2Ring can you make the aperture of the laser to be less that the diameter of the earth?  Something realistic, say the diameter of the Hubble telescope (2.4 m)
D Duck
CC BY-SA 4.0
2024-12-23T20:20:56.637Z
The answer is good, except one little detail: the 46 Glyr for the observable universe is the distance a laser light emitted at the big bang could travel until today, but if the question is about the comoving distance a laser light emitted today could travel that is the event horizon at 17 Glyr. If you take the particle horizon the universe was denser at the time of the big bang when the light was emitted and the likelyhood to hit something was therefore also higher.
Yukterez
CC BY-SA 4.0
2024-12-23T22:33:14.917Z
@Yuketerez as I said, "A better calculation... would take the expansion of the universe into account".
ProfRob
CC BY-SA 4.0
2024-12-24T18:14:52.463Z
If those "Stars" calculations are true, then why did the Hubble Deep Field image see soooooo many stars in a tiny, allegedly empty area of space?
RonJohn
CC BY-SA 4.0
2024-12-24T19:13:59.137Z
@RonJohn because it wasn't a tiny area of space and there are hardly any identifiable stars in the HDF. There are lots and lots of galaxies, mostly filled with empty space. Also stars are point-like, even though the optics of the HST blurs them into pointy blobs.
ProfRob
CC BY-SA 4.0
2024-12-22T09:33:28.857Z
2024-12-22T20:58:56.617Z
2024-12-22T20:58:56.617Z
ProfRob
<p>In <a href="https://what-if.xkcd.com/109/" rel="nofollow noreferrer">this What If</a>, Randall answers what would happen if we shine a laser in a random direction.</p>
<p>When it comes to hitting a star, he says only that it is near impossible to hit one in our galaxy, even if aiming.</p>
<p>I want to know though, if the beam was completely confined (zero divergence) and the photons went on to the visible edge of the universe, what, then, are the odds of it hitting a star?</p>
<p>Same question for other objects. If it's unlikely to hit a star, what about a galaxy or nebula? If likely*, what about a planet?</p>
<p>*(I doubt it is likely)</p>

CC BY-SA 4.0
2024-12-22T21:24:00.233Z
This sounds like the inverse of [Olbers' Paradox](https://en.wikipedia.org/wiki/Olbers%27s_paradox)
Barmar
CC BY-SA 4.0
2024-12-23T15:54:19.033Z
One interesting bit of context to add, when we measure the distance to the moon using lasers, we have to aim ahead of the target mirrors.
David S
CC BY-SA 4.0
2024-12-24T23:27:01.080Z
https://physics.stackexchange.com/questions/706189/if-i-moved-in-a-straight-line-forever-would-i-hit-something?noredirect=1&lq=1

https://physics.stackexchange.com/questions/388802/if-i-sliced-the-universe-in-half-would-the-slice-go-through-a-star
htmlcoderexe
CC BY-SA 4.0
2024-12-25T06:31:23.280Z
@Barmar the Olbers' Paradox is mentioned in the WhatIf XKCD linked in the question
justhalf
CC BY-SA 4.0
2024-12-25T11:11:33.630Z
The question should clarify whether it means an idealized laser with zero divergence (so that beam width stays constant to infinity), or a practical laser that always has some amount of divergence (in which case the answer becomes trivial).
jpa
CC BY-SA 4.0
2024-12-25T11:29:27.437Z
@jpa it does: "if the beam was completely confined"
Rabbi Kaii
CC BY-SA 4.0
2024-12-25T11:51:48.423Z
@RabbiKaii Ah, ok, I didn't understand it that way :)
jpa
CC BY-SA 4.0
2024-12-25T20:42:01.283Z
@jpa I made an edit that should make OP less ambiguous
Rabbi Kaii
<star><distances>
Odds of hitting a star with a "laser" shone in a random direction
Rabbi Kaii
# Odds of hitting a star with a "laser" shone in a random direction

(Asked by: username_0 on 2024-12-22)

In [this What If](https://what-if.xkcd.com/109/), Randall answers what would happen if we shine a laser in a random direction.

When it comes to hitting a star, he says only that it is near impossible to hit one in our galaxy, even if aiming.

I want to know though, if the beam was completely confined (zero divergence) and the photons went on to the visible edge of the universe, what, then, are the odds of it hitting a star?

Same question for other objects. If it's unlikely to hit a star, what about a galaxy or nebula? If likely\*, what about a planet?

\*(I doubt it is likely)

**Tags:** star, distances
> **Comments:**
> **username_1**: This sounds like the inverse of [Olbers' Paradox](https://en.wikipedia.org/wiki/Olbers%27s_paradox)  
> **username_2**: One interesting bit of context to add, when we measure the distance to the moon using lasers, we have to aim ahead of the target mirrors.  
> **username_3**: https://physics.stackexchange.com/questions/706189/if-i-moved-in-a-straight-line-forever-would-i-hit-something?noredirect=1&lq=1  https://physics.stackexchange.com/questions/388802/if-i-sliced-the-universe-in-half-would-the-slice-go-through-a-star  
> **username_4**: @Barmar the Olbers' Paradox is mentioned in the WhatIf XKCD linked in the question  
> **username_5**: The question should clarify whether it means an idealized laser with zero divergence (so that beam width stays constant to infinity), or a practical laser that always has some amount of divergence (in which case the answer becomes trivial).  
> **username_0**: @jpa it does: "if the beam was completely confined"  
> **username_5**: @RabbiKaii Ah, ok, I didn't understand it that way :)  
> **username_0**: @jpa I made an edit that should make OP less ambiguous  

---

## 1 answers

### Answer by username_6 on 2024-12-22. Score: 23 | Accepted answer

**Stars**

One way of doing this is to take an estimate of the total number of stars in the observable universe, divide that by the volume of the observable universe to get a density, then multiply by some average cross-sectional area for a star. The reciprocal of this would be a typical path length for a random sight line to intercept the surface of a star. Note that this is a static calculation which assumes the "beam" of radiation is a pulse short enough that stellar motions do not increase their effective cross-sectional area.

e.g., Using the estimates presented [here](https://en.wikipedia.org/wiki/Observable_universe), the number of stars in the (observable) universe is $N\sim 10^{24}$, the radius of the observable universe is $\sim 46.5$ billion light years, corresponding to a current volume of $V = 3.6 \times 10^{80}$ m$^3$.

The next part is tricky. The average star is a low mass M dwarf with a radius of perhaps 30% that of the Sun. But a few percent of stars are giants and much larger - maybe 100 times the radius of the Sun and I think these will dominate in terms of average cross-sectional area. I.e.
$$\bar{\sigma} \sim \left( \frac{97 \times 0.3^2 + 3\times 100^2}{100}\right) \pi R\_\odot^2 = 5\times 10^{20}\ {\rm m}^2\ .$$

Putting these together, the mean free path of a photon beam through the universe, before intercepting the surface of a star is
$$ l = \frac{V}{N\bar{\sigma}}\sim 10^{20}\ {\rm light\ years}$$
and the chance of a random beam of light travelling the radius of the observable universe intercepting with a star is $\sim 46.5\times 10^9/10^{20}\sim 5\times10^{-9}$.

This can only be an order of magnitude estimate. A better calculation could make a better demographic survey of stellar sizes (as a function of cosmic epoch), would take the expansion of the universe into account (in the future, the average stellar density would be lower) and that stars are not uniformly distributed, but clumped into galaxies. It also ignores the Sun, which contributes an interception probability of $5\times 10^{-6}$ to a beam launched from Earth.

**Planets**

Even if we allow each star to have 10 planets, each planet is say 10-1000 times smaller than its star so contributes a fraction of the cross-sectional area. So the chances of light intercepting a planet are orders of magnitude lower than for stars.

(Note this ignores the trivial fact that a randomly directed beam launched from Earth has a 50% chance of hitting the Earth and a $5\times 10^{-6}$ chance of hitting the Moon.)

**Galaxies**

If we say our $10^{24}$ stars are actually in about $10^{14}$ galaxies, each with a radius of about 30 thousand light years (a bit smaller than the Milky Way), a similar calculation gives a mean free path of about a billion light years.

**Thus the overall conclusion is that a random light beam is likely to pass through many galaxies, but highly unlikely ever to intercept a star.**

> **Comments:**
> **username_7**: Your laser beam is a mathematical line, right? With no cross-sectional area or divergence (or extinction). What are the probabilities like if we use a more realistic beam?  
> **username_8**: Shouldn't the duration of beaming be taken into account? For instance if aiming a random spot on cmb, stars that have been missed could cross the continuous line later. Also if beam originates say from equator in a direction aligned with equatorial plane, there's 100% chances the beam crosses the Earth in the next ~12h  
> **username_6**: @user721108 Yes, I'm sure that's right. You could make the answer much more complex. My calculation assumes an outwardly directed pulse.  
> **username_9**: @PM2Ring can you make the aperture of the laser to be less that the diameter of the earth?  Something realistic, say the diameter of the Hubble telescope (2.4 m)  
> **username_10**: The answer is good, except one little detail: the 46 Glyr for the observable universe is the distance a laser light emitted at the big bang could travel until today, but if the question is about the comoving distance a laser light emitted today could travel that is the event horizon at 17 Glyr. If you take the particle horizon the universe was denser at the time of the big bang when the light was emitted and the likelyhood to hit something was therefore also higher.  
> **username_6**: @Yuketerez as I said, "A better calculation... would take the expansion of the universe into account".  
> **username_11**: If those "Stars" calculations are true, then why did the Hubble Deep Field image see soooooo many stars in a tiny, allegedly empty area of space?  
> **username_6**: @RonJohn because it wasn't a tiny area of space and there are hardly any identifiable stars in the HDF. There are lots and lots of galaxies, mostly filled with empty space. Also stars are point-like, even though the optics of the HST blurs them into pointy blobs.  

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