This week’s post is a bit of a distraction from the usual business, based on a question I wondered about. Rather than ask Google, I dug in like a nerd to get a more complete picture.

One of my frequent spiels is about the vastness of space, in the context that we can dismiss fantasies about humans traveling to the stars. I do throw in an old-school calculation at the end to reinforce this point, but until then we’ll entertain ourselves with a sense for the scale of the sky we see with our eyes.

When we consider a scale model in which the sun is reduced to the size of a sand grain (about 1 millimeter), the closest neighbor star is about 30 km away. One light year at this scale is about 7 km. But how typical is this yawning gulf in our region of the galaxy? And how far away are the stars we lay eyes on in the night sky, typically?

Before getting to those questions, just how many stars can we see, naked-eye? It depends on the darkness of your sky. According to the Hipparcos catalog, rounding apparent visual magnitudes to the nearest integer, there are two −1 magnitude stars: Sirius and Canopus. Eight more join at magnitude zero; 12 at first magnitude; 71 at second; 192 at third; 622 at fourth; 1909 at fifth; and 5976 at sixth—at which point our eyes run out of steam. A suburban sky might allow fourth magnitude, or roughly 1,000 stars (not all at once, since only half are up at a time). At fifth magnitude, we get about 3,000 (all-sky). At the limit, we tally about 9,000 stars. About half this number would be above the horizon at any given time.

Incidentally, going to space hardly does a thing to improve visibility: the atmosphere is pretty impressively transparent at visible wavelengths (only “eating” about a tenth of a magnitude). I was excited to see the night sky from Mauna Kea on my first observing trip there as a graduate student. Being above 40% of the Earth’s atmosphere, it’s the closest I had been to space. The thing is, low oxygen levels impair visual sensitivity, so when I first went outside it really sucked: I could barely see a thing (eventually dark-adapted, but way slower than at lower elevations). Space is even worse on the oxygen front.

Distances the Dumb Way

The fact that we can see a few thousand stars in the sky crudely implies that we see a box of stars spanning about 20 star separations on a side. I’m sure I lost most people, there. What I mean is if we took the simplest possible model, giving each star a cubic domain and arranging them all in a cubic lattice, a 20×20×20 set of such “star boxes” would total 8,000 stars—roughly matching what we can see naked-eye (10 by 10 by 10 would be shy at 1,000 stars). The model is too stupid to bother nailing any particular number. In any case, sitting at the middle of this box, we would see 10 stars out in each direction, which would be about 45 light years if we use the closest star at 4.5 light years to set the typical box spacing.

That’s it? just 50 light years away? Well, not really. Stars come in all kinds of masses (thus luminosities), so we can’t see many of the closest stars because they’re too dim, while we see very distant stars that blaze away.

Enter Hipparcos

The Hipparcos mission performed careful measurements of over 100,000 star positions, measuring the parallax of all stars in the sky to well below the naked-eye limit. Parallax is the apparent annual wiggle of star positions due to Earth’s orbit around the sun; the amount of wiggle decreasing for farther stars. The astronomical distance called the parsec (3.26 light years) derives from parallax observations. A star at a distance of one parsec (none are quite that close) would have a wiggle amplitude on the sky of one arcsecond. These wobbles are thus very very small (well less than an arcsecond), and a challenge to measure. But that’s what Hipparcos did.

Using the original release of the Hipparcos catalog I studied the “real” story on a Sunday evening two weeks back. The short answer is that almost no matter the visual magnitude (brightness of the star), typical distances are tens to hundreds of light years.

Rather than provide a mind-numbing table, what follows is a more digestible walk through the story from the brightest to the dimmest stars we see in the sky.  A quick word about magnitudes.  The magnitude scale is logarithmic, where lower equals brighter, and in which a difference of five magnitudes translates to a factor of 100 in brightness.  Apparent magnitudes are what we usually mean when we speak of a star’s magnitude: how it looks at the distance it is.  The same kind of star has a lower apparent magnitude (brighter) the closer it is.  Absolute magnitudes level the playing field, asking what the apparent magnitude would be if viewing the star from a distance of 10 parsecs (32.6 light years).  It gets at the intrinsic brightness of the star.  For instance, the sun is intrinsically dimmer than almost any star we see when we look up to the sky, at absolute magnitude M = 4.8.  Yet, its extreme proximity gives it an apparent magnitude of m = −26.7 (a factor of 4×10¹² owing to the sun being 2×10⁶ times closer than 10 parsecs).  The convention uses a lower-case m for apparent magnitude and upper case for absolute.

The Two Brightest

Sirius, at m = −1.44 mag and Canopus at m = −0.62 top the list. They are, respectively, 8.6 and 310 light years away. Already, this is remarkably typical of the ranges we’ll find at almost every brightness. The disparity is also an early lesson: the variations are large and untidy.

Zero Magnitude

We pick up 8 stars here, whose distances range from 4.4 light years (Alpha Centauri A) to 773 light years (Rigel). Ignoring the extremes (a practice we’ll keep doing), we get the range from 11.4 to 428 light years (Procyon to Betelgeuse). Either way, Orion wins for the distant end at zeroth magnitude. The median distance is 40 light years.

First Magnitude

Twelve more stars come on board. The closest is again Alpha Centauri (the B component of a double) at 4.4 light years, and the farthest is Deneb, which is at 1,550 light years. Excluding these two extremes leaves 16.8 to 604 light years (Altair and Antares). Deneb deserves special attention here. It’s the most distant bright star in the sky, and member of the prominent Summer Triangle. At an absolute magnitude of M = −7.1, it’s about as intrinsically bright as stars come.  Only two second-magnitude stars are farther: eta Canis Major (tail of the dog), at 3,200 light years (m = 2.5 magnitude), and delta Canis Major (its rump), at 1790 light years (m = 1.83 mag). Then only five more (of 192) third-magnitude stars are farther than Deneb. So it stands out. The median distance for first magnitude stars is 170 light years.

Second Magnitude

We add 71 stars now, with a median distance of 177 light years (almost the same as for the previous group of 12). The range is 36 to 3,200 light years. Excluding the closest 10% (7) and farthest 10% (7) to capture typical stars—not outliers—we’re left with 66 to 820 light years.

Third Magnitude

192 new arrivals produce a median of 206 light years, ranging from 12 to 2,570 (note, not as far as the farthest second-magnitude stars). Stripping off the closest 10% and farthest 10% leaves the range from 60 to 730 light years. Again, pretty similar to second magnitude results. Through third magnitude (loosely speaking, those that form the recognizable constellations), only 18 of the 285 stars (about a sixteenth) are farther than 1,000 light years.

Fourth Magnitude

Now we get 622 additions having a median distance of 230 light years, ranging from 10.5 to 8,800 light years. The central 80% runs from 70 to 965 light years.

Stars dimmer than fourth magnitude don’t stand out, and require care to notice in most sky conditions. So, we can pause here to reflect on the fact that aside from a few outliers like Deneb and eta/delta Canis Major, you’re unlikely to randomly point to a “bright” star more distant than 1,000 light years (I disproved this to myself in a hardware experiment briefly described below!). Aside from Alpha Centauri and Sirius, you’re not below 10. Typical distances are a few hundred light years.

Fifth Magnitude

Now we get into dim star territory, finding a median distance of 350 light years, a min-to-max of 11 to 325,000 (max number Hipparcos will assign, at parallax sensitivity limit), and a central 80% ranging from 110 to 1,075. Still basically saying: hundreds of light years, typically. I would be disinclined to take the 325,000 light year numbers seriously—possibly just indicating a failure to measure a reliable parallax.

Sixth Magnitude

At the limit of our visual perception (naked eye), and only in the darkest sites on clear nights without a moon, we get to a median of 450 light years and a min-to-max the same as for fifth magnitude. The central 80% occupies 150 to 1,280 light years, which is still saying: hundreds.

Synthesis

Even out to the limit of our vision, 90% of the stars we might point to—even at the dimmest end—are closer than roughly 1,000 light years. This is only 1% the scale of our galaxy, emphasizing the very local nature of our experience when looking at the stars. The following table counts how many stars are farther than 1,000 light years (out of how many in each magnitude bin), computing the percentage beyond 1,000 light years—both within the bin and cumulatively up to that magnitude.

Mag	>1,000 l.y.	out of	%	cumulative %
−1	0	2	0	0
0	0	8	0	0
1	1	12	8.3	4.6
2	5	71	7.0	6.5
3	12	192	6.3	6.3
4	59	622	9.5	8.5
5	210	1909	11.0	10.2
6	943	5967	15.8	14.0

Theory Comparison

While the exercise above using the Hipparcos catalog satisfied my initial curiosity, I couldn’t help but wonder about theoretical expectations, typical distances between stars, etc. So, I found a paper from 1982 with an analytic fit to the “luminosity function” of local stars (Figure 5 in paper). That is to say: how many stars per cubic parsec in each intrinsic (absolute) magnitude bin? Really bright stars are rare, but can be seen far away. Dim jobs are a million times more abundant, but often too dim to see.

Armed with this knowledge, one can build a theoretical expectation for how many stars one might see at various apparent brightnesses. For example, second magnitude stars encompass the range of m = 1.5 to 2.5. A brightest-of-the-bright star with absolute magnitude of M = −8 (about 100,000 times more luminous than our sun, whose absolute magnitude, recall, is M = 4.8) would appear as a second magnitude star at a distance of 800 to 1,250 parsecs (2,500 to 4,100 light years), encompassing a 3-D volume of 6 billion cubic parsecs. Exceedingly rare, the luminosity function predicts 5×10⁻¹⁰ of stars this bright for every cubic parsec: very low odds in any given cubic parsec. But we have lots of volume, ending up with an expectation of three such M = −8 stars appearing as a second-magnitude star (in actual fact, we see only one).

This game can be played for every intrinsic (absolute) magnitude and every apparent magnitude to produce the following plot.

Each box corresponds to an apparent magnitude (what we see on the sky). Within each are theoretical expectations as horizontal bars starting at absolute magnitude M = −8 on the right (can see far away) and working toward dimmer stars as one comes left. In the m = 2 case, the right-most bar corresponds to the example in the paragraph above: spans 2,500–4,100 light years, at a level of 3.

Star symbols represent the Hipparcos data (counts per corresponding distance bin in theoretical “curve”). Note Sirius and Canopus in the m = −1 box, each as somewhat unlikely outliers in the distribution. Deneb is also visible as the rightmost star in the m = 1 box. Theory expected one, and we got one.

Labels are provided for absolute magnitudes of M = −8, 0, and 5 (like our sun). Notice that we don’t expect to see many stars like our sun (too dim unless very close). We got lucky in having Alpha Centauri right next door: wasn’t particularly likely, at a few-percent chance.

As we progress from one box to the next in the figure, the number of observed stars initially tracks the theoretical expectation reasonably well (large relative Poisson error on small-number cases), but soon the actual counts fall well short (compare “expected” to actually “got” numbers in the boxes). Theory and experiment line up reasonably well on the left side of the plots, but begin to dive dramatically at larger distances—starting at a few hundred light years. At first, I suspected that Hipparcos simply had trouble measuring parallax to more distant stars, and that inclusion in the catalog tapered as one went farther out. But no: Hipparcos is complete up to at least 7.3 magnitude, so the naked-eye stars are all there.

What gives? My educated guess is two things that work in parallel. Perpendicular to the disk of the galaxy, we run out of stars. The disk is about 1,000 light years thick (not a hard edge). In the plane of the galaxy, intervening gas and dust obscures more distant sight-lines. In both cases, our window to the stars is truncated, making it somewhat rare to see stars beyond 1,000 light years.

The Unseen Dim

The luminosity function indicates a monotonically increasing population density as stars get less massive and therefore less intrinsically bright. Yet the stars we’re most likely to see are at the brighter end. In fact, brighter than apparent magnitude 4.5, only seven stars in the sky are intrinsically dimmer than the sun. One happens to be Alpha Centauri (B), at 4.4 light years. Only two others are brighter than fourth magnitude, and both near each other on the sky, visible at most latitudes on Earth: tau Cetus is a G8 dwarf star (our sun is a G2 dwarf) 12 light years away with absolute magnitude 5.7, and epsilon Eridanus is a K2 dwarf 10.5 light years away at absolute magnitude 6.2. The dimmest two stars potentially visible to the naked eye are actually in a somewhat famous binary pair (appear as a single star to the eye): a K5 and K7 dwarf 11.4 light years away called 61 Cygnus with a combined magnitude of 5. In a decent sky, you can see it naked-eye, and know that you won’t see anything intrinsically dimmer in the sky (by eye), having absolute magnitudes of 7.5 and 8.3.

Even though stars dimmer than this are the most abundant in the galaxy, none are close enough to make out naked-eye.

Typical Interstellar Distances

Using the luminosity function for stars, which shows increased abundance for smaller/dimmer stars, we can examine the space density at each absolute magnitude to determine how far such stars likely are from each other (the cube-root of how large a volume each tends to occupy). The table below has, for each absolute magnitude, the typical distance between stars of that intrinsic brightness and also the distance between all stars this bright or brighter. For instance, the sun has an absolute magnitude of M = 4.8 (call it 5). On average, such stars might be found every 22 light years, and you’d go 16 light years on average before hitting any star as (intrinsically) bright or brighter.

Abs. Mag	Avg. Dist. (l.y.)	Cumulative
−8	>4040	4040
−7	2300	2180
−6	1330	1240
−5	770	715
−4	450	420
−3	270	250
−2	165	150
−1	105	95
0	70	62
1	49	43
2	37	31
3	29	24
4	25	19
5	22	16
6	20	14
7	18.5	12.4
8	17	11.2
9	16.6	10.3
10	16	9.5
11	15.4	8.8
12	14.8	8.3
13	14.4	7.8
14	13.9	7.4
15	13.5	7

Even the super-abundant dim-@$$ stars at the bottom of the list tend to be over ten light years apart, and the cumulative effect is stars of any size averaging 7 light years apart. So it’s a bit of a fluke that we have nearly-twin stars less than 5 light yeas away, and Sirius (absolute magnitude M = 1.43) within 10 light years (expect over 30 light years before finding one this intrinsically bright).

This makes me rethink my grains of sand 30 km apart, which was based on the distance to Alpha Centauri. If counting stars as bright as our sun or brighter, 16 light years means these grains are typically over 100 km apart. Even including the merest specks of barely-stars, typical interstellar distances put sand grains 50 km apart on this scale.

Side Hustle

In exploring this issue, I realized I’d like to be able to point to a star in the actual sky and know how far it is. Then I realized I had relevant equipment (encoders, reflex sight, raspberry pi) and knowledge (spherical astronomy, catalogs, interface/programming) to whip something up. I just tested it the night before posting this, and it works well: put the red dot on a star and the screen gives its stats: name, apparent magnitude, distance, spectral type. I stumbled on one third-magnitude star (epsilon Auriga) over 2,000 light years away. Mostly it was hundreds—like the post said, at some length.

Inaccessible

I don’t want to undercut my usual message of how vast and empty space is by making it sound like all the stars we see are sort-of close. The main point to take away is: tens and hundreds of light years—that’s what we typically see. But just to be sure, here’s a fun calculation showing that even 5 light yeas is crazy far in nay practical sense.

If using the only sort of propulsion we’ve ever used to move humans through space (chemical rockets), let’s see how long it would take to reach the nearest star, just over four light years away. We’ll pack minimally, and try to get away with a 10 ton payload—four times less than the payload delivered to the moon by Saturn V rockets (don’t ask me how to pare down to this: forget the toothbrush!). We’ll also use a fuel mass equivalent to the entire fossil fuel endowment of Earth: let’s say five trillion barrels of oil equivalent, coming to a mass of 500 billion tons. We’re not messing around, here. Go big or stay home!

The logarithmic rocket equation using a high exhaust velocity of 4,000 meters per second would produce a payload velocity of about 100,000 meters per second, or one-three-thousandth the speed of light. Therefore, it would take over 12,000 years to reach the nearest star. Don’t wait up. This is using the entire fossil fuel provision of Earth (or its mass equivalent), 50 billion times the mass of the payload. Logarithms are cruel, so that nuances to this crude calculation won’t change the overall conclusion. Guess we’ll stay home, then.

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