<p>Take a look at Tokovinin (2014) <a href="https://browse.arxiv.org/abs/1401.6827" rel="nofollow noreferrer">From binaries to multiples II: hierarchical multiplicity of F and G dwarfs</a> for a discussion of the statistics of binary pairs and the issues in estimating things like the average distance.</p>
<p>The distribution will be highly skewed, so the mean may not give a good measure of central tendency. The sample will be strongly affected by observation bias.  It is far easier to observe wide binary pairs than close ones. For many spectroscopic binaries the long term study needed to determine orbital parameters hasn't been done. These mean that there are considerable unknown uncertainties.</p>
<p>That said, the paper does report that the distribution of orbital periods follows a log normal distribution with a median of about 100 years. Given a reasonable value for the mass of the stars in a binary pair, this corresponds to a separation of about 16 AU.  The mean would presumably be larger than this, given the skew in the distribution.</p>

CC BY-SA 4.0
2023-10-09T03:33:20.063Z
Yeah there's always issues with distribution and biases. Like imagine if the sample size is 100, and the first half have a value of 3, and the last half have a value of 80. Then even taking the **median** is not gonna give a good idea of the "average" as you prolly imagine it. However there is one bias I'm concerned about more than others: `Observation bias. It is far easier to observe wide binary pairs than close ones.` Can you at least give me a concrete number for the difference between "wide" and "narrow"? Where is that threshold? 5 AU? 10 AU? Something else?
DrZ214
CC BY-SA 4.0
2023-10-09T05:42:17.303Z
I think it says in that paper that "32% of binaries with a period of less than 100 yrs are known only spectroscopically and have no orbit information (other than "close")
James K
CC BY-SA 4.0
2023-10-09T05:45:43.647Z
conversely some of the "wide binaries" are probably not binaries at all, but just two stars that happen to align.
James K
CC BY-SA 4.0
2023-10-10T06:39:14.017Z
F and G stars are not average stars. The overall distribution will be determined by M dwarfs.
ProfRob
CC BY-SA 4.0
2023-10-08T23:00:13.240Z
2023-10-09T22:46:07.313Z
2023-10-09T22:46:07.313Z
James K
<p>Note I'm exclusively considering binary stars, not 3+ star systems.</p>
<p>Binary stars come in all extremes. <a href="https://en.wikipedia.org/wiki/Binary_star#Orbital_period" rel="nofollow noreferrer">Wikipedia</a> says that their orbital periods can be a few hours, a few days, or hundreds of thousands of years.</p>
<p>Is there any data, however, on the average semi-major axis of their orbits? I'm trying to get an idea of how far apart the stars are from each other, and would like an average across all measured binaries in the Milky Way.</p>

CC BY-SA 4.0
2023-10-08T13:42:07.897Z
What sort of average:-  mean, median? The trouble with mean would be that the distribution of semi major axes would be highly skewed with a long tail, so the mean might not be the best or most stable.
James K
CC BY-SA 4.0
2023-10-08T16:20:24.687Z
There's also going to be lots of difficulty with sample bias.  Binary stars are easier to observe when they are widely spaced, many close binaries are only known to be binaries by spectroscopy. orbital determination of these requires extensive follow-up observations, and hence won't be done for many such stars.  This results in observational bias But see https://browse.arxiv.org/pdf/1401.6827.pdf (and other works of that author) for more about the statistics.  But let me throw a number out:  16AU: from "lognormal distribution of period with median 100yr" and accounting for lower mean mass.
James K
CC BY-SA 4.0
2023-10-08T21:06:55.800Z
@JamesK I think your comment with the reference to this paper is worth an answer - even without much or any expansion. It has the direct answer to the question asked as quote from the paper's §5 summary - and it shows, that the question is not well-posed due to the nature of the distribution and the answer will mis-lead, and shows how one can do better (by following what the author of the paper does). Regardless: thanks a lot for making me aware of that very interesting paper!
planetmaker
CC BY-SA 4.0
2023-10-08T23:08:29.167Z
The answer is mass-dependent.
ProfRob
CC BY-SA 4.0
2023-10-09T22:43:57.767Z
different but related: [What is a thermal distribution of eccentricities?](https://astronomy.stackexchange.com/q/34535/7982)
uhoh
<orbit><binary-star>
For binary stars, what is the average semi-major axis?
DrZ214
# For binary stars, what is the average semi-major axis?

(Asked by: username_0 on 2023-10-08)

Note I'm exclusively considering binary stars, not 3+ star systems.

Binary stars come in all extremes. [Wikipedia](https://en.wikipedia.org/wiki/Binary_star#Orbital_period) says that their orbital periods can be a few hours, a few days, or hundreds of thousands of years.

Is there any data, however, on the average semi-major axis of their orbits? I'm trying to get an idea of how far apart the stars are from each other, and would like an average across all measured binaries in the Milky Way.

**Tags:** orbit, binary-star
> **Comments:**
> **username_1**: What sort of average:-  mean, median? The trouble with mean would be that the distribution of semi major axes would be highly skewed with a long tail, so the mean might not be the best or most stable.  
> **username_1**: There's also going to be lots of difficulty with sample bias.  Binary stars are easier to observe when they are widely spaced, many close binaries are only known to be binaries by spectroscopy. orbital determination of these requires extensive follow-up observations, and hence won't be done for many such stars.  This results in observational bias But see https://browse.arxiv.org/pdf/1401.6827.pdf (and other works of that author) for more about the statistics.  But let me throw a number out:  16AU: from "lognormal distribution of period with median 100yr" and accounting for lower mean mass.  
> **username_2**: @JamesK I think your comment with the reference to this paper is worth an answer - even without much or any expansion. It has the direct answer to the question asked as quote from the paper's §5 summary - and it shows, that the question is not well-posed due to the nature of the distribution and the answer will mis-lead, and shows how one can do better (by following what the author of the paper does). Regardless: thanks a lot for making me aware of that very interesting paper!  
> **username_3**: The answer is mass-dependent.  
> **username_4**: different but related: [What is a thermal distribution of eccentricities?](https://astronomy.stackexchange.com/q/34535/7982)  

---

## 1 answers

### Answer by username_1 on 2023-10-08. Score: 4

Take a look at Tokovinin (2014) [From binaries to multiples II: hierarchical multiplicity of F and G dwarfs](https://browse.arxiv.org/abs/1401.6827) for a discussion of the statistics of binary pairs and the issues in estimating things like the average distance.

The distribution will be highly skewed, so the mean may not give a good measure of central tendency. The sample will be strongly affected by observation bias. It is far easier to observe wide binary pairs than close ones. For many spectroscopic binaries the long term study needed to determine orbital parameters hasn't been done. These mean that there are considerable unknown uncertainties.

That said, the paper does report that the distribution of orbital periods follows a log normal distribution with a median of about 100 years. Given a reasonable value for the mass of the stars in a binary pair, this corresponds to a separation of about 16 AU. The mean would presumably be larger than this, given the skew in the distribution.

> **Comments:**
> **username_0**: Yeah there's always issues with distribution and biases. Like imagine if the sample size is 100, and the first half have a value of 3, and the last half have a value of 80. Then even taking the **median** is not gonna give a good idea of the "average" as you prolly imagine it. However there is one bias I'm concerned about more than others: `Observation bias. It is far easier to observe wide binary pairs than close ones.` Can you at least give me a concrete number for the difference between "wide" and "narrow"? Where is that threshold? 5 AU? 10 AU? Something else?  
> **username_1**: I think it says in that paper that "32% of binaries with a period of less than 100 yrs are known only spectroscopically and have no orbit information (other than "close")  
> **username_1**: conversely some of the "wide binaries" are probably not binaries at all, but just two stars that happen to align.  
> **username_3**: F and G stars are not average stars. The overall distribution will be determined by M dwarfs.  

---

