<p>What you are looking for is called the &quot;galactic size-mass relation&quot; for galaxies dominated by disks.</p>
<p>Theres an <a href="https://watermark.silverchair.com/stu2467.pdf?token=AQECAHi208BE49Ooan9kkhW_Ercy7Dm3ZL_9Cf3qfKAc485ysgAAAqQwggKgBgkqhkiG9w0BBwagggKRMIICjQIBADCCAoYGCSqGSIb3DQEHATAeBglghkgBZQMEAS4wEQQMKmKveevXpP_senyYAgEQgIICVx79w85CKSYxDJljIuKMW31P_iAqWRFbZsT2K3hkZ_PShO2Kk9_GiWXODFMnRnB1u0KhF0msKU0OATwRAKz61Wp_hBSOPDxSxOr_HS1-bZJdL3hpMntytL-SmYU_fSUllpO8GIHk3nlcT__LuWn4xKj_raFzvvvlujD9_vNWaRYAOLmNQjnZsN5jnBQD8NcQIS4mYckYY-Itjt99kIrKLuItOwaOhzWz0JasVTR-KaHfoW5_ClxlYBpAH8Dzl8r6jDF-ssl80SAuRkCehyF2VQ1Izx6qyC0ctMN-1rM563NMmaaBMYPgB9NrKt_g2YCKCEY4iGJIuHPdLsbWX4w2IMBcj97UzWnt0prj2HE6EGcy-7knE4XXK3op4NcJ8x-_0bDBiVnXT4ca0zy_XurwzQpiZeTIwNSWZ0pv4Zl59eq1h7GaNbYjfRy5hG7LE1lXJa_RoMtEHeVUCs4GHp7ORVFX9m0sj_ClbI8TLY7SartHArO8jc6kSCkH6FBVAudwDmSn3hRfYPAuI-bOt3gnsBak4-z7nicgCrVQv4LqxrG57KI_LcsRE5eA7B9YQ-R1bnHjuTcsRsPYZFoIkSDeTmzc9JdoHiRzN11212jMcA6aix9eIsQMHKzdP7kJf0uaehRG9YepoZT2FyOHp0K2DB7XFHpJpFM2zhTR79X_PZQBD3dOSQcz2TZZls9sgT_qIAbPXlkuac1O0CKpZcLVg6VogzA4MevWBzUVn7I6zWrTzv50nrRm2qhpZfAG6E7cPDaBw4rwiDSek16Xt9zew9_1-Efgl7uQ" rel="nofollow noreferrer">interesting research paper by Rebeca Lange</a> and others where this relation (equation 3 in the paper) has the form</p>
<p><span class="math-container">$R = \gamma M^{\alpha}(1+M/M_0)^{\beta-\alpha}$</span></p>
<p>Where <span class="math-container">$R$</span> is the radius of the galaxy in kiloparsecs (<span class="math-container">$1$</span> kpc <span class="math-container">$= 3260$</span> light year), <span class="math-container">$M$</span> is the mass of the galaxy measured in solar masses (<span class="math-container">$1$</span> solar mass <span class="math-container">$= 1.988\cdot 10^{30}\; kg$</span>), and <span class="math-container">$\gamma$</span>, <span class="math-container">$\alpha$</span>, <span class="math-container">$\beta$</span> and <span class="math-container">$M_0$</span> are parameters that have been tweaked to adjust the relation to the real data.</p>
<p>In their paper they propose values for <span class="math-container">$\gamma$</span>, <span class="math-container">$\alpha$</span>, <span class="math-container">$\beta$</span> and <span class="math-container">$M_0$</span> for different kinds of measurements and galaxies. For the late-type galaxies (which are the ones that you want to study), these parameters are shown in table 2 with values that have a certain range.</p>
<ul>
<li><span class="math-container">$\alpha$</span> between <span class="math-container">$0.13$</span> and <span class="math-container">$0.17$</span></li>
<li><span class="math-container">$\beta$</span> between <span class="math-container">$0.91$</span> and <span class="math-container">$1.00$</span></li>
<li><span class="math-container">$\gamma$</span> between <span class="math-container">$0.08$</span> and <span class="math-container">$0.18$</span></li>
<li><span class="math-container">$M_0$</span> between <span class="math-container">$19\cdot 10^{10}$</span> and <span class="math-container">$48\cdot 10^{10}$</span> solar masses</li>
</ul>
<p>To make an approximate guess let's take the average values they present for each parameter. Then you will have the relation:</p>
<p><span class="math-container">$R = 0.12\cdot M^{0.15}\cdot(1+3.8\cdot 10^{-12}M)^{0.81}$</span></p>
<p>Again with <span class="math-container">$R$</span> in <span class="math-container">$kpc$</span> and <span class="math-container">$M$</span> in solar masses.</p>
<p>With everything into account we are ready to plug-in some numbers. For example, the Milky Way galaxy is known to have a mass of <span class="math-container">$M = 10^{12}$</span> solar masses. This means that according to our model the Milky Way disk has a radius of <span class="math-container">$R = 26.97 \; kpc$</span>. Well it turns out that we think the disk to extend between <span class="math-container">$170,000$</span> and <span class="math-container">$200,000$</span> light years in diameter, which translates to <span class="math-container">$26 \; kpc$</span> to <span class="math-container">$31\; kpc$</span>, so we proved that the model works at least for our galaxy, even if our prediction is in the lower bound of the range.</p>
<hr />
<p>Now, the devil is in the details; <span class="math-container">$R$</span> is not actually the radius of the disk as you might think. The radius is an ambiguous term here since disk galaxies don't have a sharp edge where stars stop to exist. The <span class="math-container">$R$</span> in the formula is actually the so-called <a href="https://en.wikipedia.org/wiki/Effective_radius" rel="nofollow noreferrer">&quot;effective radius&quot;</a>, wich is in fact just the radius which contains half of the light emmited from the galaxy (this is an unambiguous way to measure the size of the galaxy). So, when talking about the radius <span class="math-container">$R$</span> I would take this into consideration, and use maybe a <span class="math-container">$1.2R$</span> estimate (20% larger) as my &quot;true&quot; radius for the galaxy.</p>
<p>Another important thing to consider is that the mass of the milky way has been measured including the mass of the dark matter and the mass of the gass. If you only want stellar mass, then the calculation will yield a galaxy that might be smaller.</p>
<p>These two are unimportant points if you are making just a highschool project and they don'tchange drastically your models either way, but I have to warn you about a consideration here. A galaxy that has a mass of <span class="math-container">$x$</span> solar masses is not a galaxy that has <span class="math-container">$x$</span> stars (planetary systems). That would certainly happen if all the stars where like the Sun, but as it turns out the vast majority of stars are less massive than the Sun, thus an <span class="math-container">$x$</span> solar mass galaxy will have much more than <span class="math-container">$x$</span> stars. If you are going to play with the Drake equiation that might be important. Finally if you search for the Milky Way data you will see that we think it has a mass of <span class="math-container">$M = 10^{12}$</span> solar masses, as I said, and we think it has around <span class="math-container">$2\cdot 10^{11}$</span> stars, so what's happening here? Shouldn't a galaxy with mass <span class="math-container">$x$</span> solar masses have more than <span class="math-container">$x$</span> stars? The problem here is, as I said previously, that we are considering the mass of the entire galaxy including gas, dust and dark matter, but we are only concerned about stellar mass.</p>
<hr />
<p>I'm very interested in your project. Would you mind to tell me more about it? I'm just curious.</p>

CC BY-SA 4.0
2020-08-12T20:03:11.677Z
thank you very much. In the research project, we will try to estimate the amount of communication between interstellar empires. First off we will make a simple stellar map of the galaxy. then randomly create alien civilizations in the habitable zone of the galaxy with a gaussian distribution (too far inside means too much chaos, too far outside means too little energy) and finally we use a sorting algorithm to figure out the distances between the different civilizations. we currently assume that civilizations within 20 light years of each other will be able to communicate
A. Soreq
CC BY-SA 4.0
2020-08-12T20:59:08.583Z
@A.Soreq That is awesome! I've been playing with similar simulations. The results are mindbending, for civilizations to contact each other you have to assume the galaxy has many many of them. The distances are crazy. I would love to hear more about this project, maybe you could PM me somehow when you have done it?
Swike
CC BY-SA 4.0
2020-08-12T21:00:07.500Z
of course. I'd love to.
A. Soreq
CC BY-SA 4.0
2020-09-01T07:51:17.630Z
https://docs.google.com/document/d/1YZuBhxe9sIYj4Biu1fLAhB1bXaZ6qoPX87GPGcSON0g/edit?usp=sharing. as promised.
A. Soreq
CC BY-SA 4.0
2020-09-01T10:21:08.663Z
@A.Soreq That is awesome! Thank you so much. And thank you for the reference to AstronomyStack also!
Swike
CC BY-SA 4.0
2020-08-12T19:55:18.080Z
2020-08-12T19:55:18.080Z
Swike
<p>I'm doing a high school programming project on drake's equation (simple galactic model) and wondered if there was a (simplified) way to find the radius (measured by the distance between the center and furthest star) of a galaxy based on its stellar mass. I found a few websites but I don't understand many of the terms and symbols in the equations shown. would love a simplified equation if possible.</p>
<p>In the project, I am modeling a 2d circular disk galaxy with no rotation.</p>
<p>I have a Y10 level understanding of maths.</p>

<galaxy><astrophysics><mathematics>
how to find the approximate [stellar radius]? (distance from center to most distant star) of a circular flat disk galaxy based on its stellar mass?
A. Soreq
# how to find the approximate [stellar radius]? (distance from center to most distant star) of a circular flat disk galaxy based on its stellar mass?

(Asked by: username_0 on 2020-08-12)

I'm doing a high school programming project on drake's equation (simple galactic model) and wondered if there was a (simplified) way to find the radius (measured by the distance between the center and furthest star) of a galaxy based on its stellar mass. I found a few websites but I don't understand many of the terms and symbols in the equations shown. would love a simplified equation if possible.

In the project, I am modeling a 2d circular disk galaxy with no rotation.

I have a Y10 level understanding of maths.

**Tags:** galaxy, astrophysics, mathematics

---

## 1 answers

### Answer by username_1 on 2020-08-12. Score: 3 | Accepted answer

What you are looking for is called the "galactic size-mass relation" for galaxies dominated by disks.

Theres an [interesting research paper by Rebeca Lange](https://watermark.silverchair.com/stu2467.pdf?token=AQECAHi208BE49Ooan9kkhW_Ercy7Dm3ZL_9Cf3qfKAc485ysgAAAqQwggKgBgkqhkiG9w0BBwagggKRMIICjQIBADCCAoYGCSqGSIb3DQEHATAeBglghkgBZQMEAS4wEQQMKmKveevXpP_senyYAgEQgIICVx79w85CKSYxDJljIuKMW31P_iAqWRFbZsT2K3hkZ_PShO2Kk9_GiWXODFMnRnB1u0KhF0msKU0OATwRAKz61Wp_hBSOPDxSxOr_HS1-bZJdL3hpMntytL-SmYU_fSUllpO8GIHk3nlcT__LuWn4xKj_raFzvvvlujD9_vNWaRYAOLmNQjnZsN5jnBQD8NcQIS4mYckYY-Itjt99kIrKLuItOwaOhzWz0JasVTR-KaHfoW5_ClxlYBpAH8Dzl8r6jDF-ssl80SAuRkCehyF2VQ1Izx6qyC0ctMN-1rM563NMmaaBMYPgB9NrKt_g2YCKCEY4iGJIuHPdLsbWX4w2IMBcj97UzWnt0prj2HE6EGcy-7knE4XXK3op4NcJ8x-_0bDBiVnXT4ca0zy_XurwzQpiZeTIwNSWZ0pv4Zl59eq1h7GaNbYjfRy5hG7LE1lXJa_RoMtEHeVUCs4GHp7ORVFX9m0sj_ClbI8TLY7SartHArO8jc6kSCkH6FBVAudwDmSn3hRfYPAuI-bOt3gnsBak4-z7nicgCrVQv4LqxrG57KI_LcsRE5eA7B9YQ-R1bnHjuTcsRsPYZFoIkSDeTmzc9JdoHiRzN11212jMcA6aix9eIsQMHKzdP7kJf0uaehRG9YepoZT2FyOHp0K2DB7XFHpJpFM2zhTR79X_PZQBD3dOSQcz2TZZls9sgT_qIAbPXlkuac1O0CKpZcLVg6VogzA4MevWBzUVn7I6zWrTzv50nrRm2qhpZfAG6E7cPDaBw4rwiDSek16Xt9zew9_1-Efgl7uQ) and others where this relation (equation 3 in the paper) has the form

$R = \gamma M^{\alpha}(1+M/M\_0)^{\beta-\alpha}$

Where $R$ is the radius of the galaxy in kiloparsecs ($1$ kpc $= 3260$ light year), $M$ is the mass of the galaxy measured in solar masses ($1$ solar mass $= 1.988\cdot 10^{30}\; kg$), and $\gamma$, $\alpha$, $\beta$ and $M\_0$ are parameters that have been tweaked to adjust the relation to the real data.

In their paper they propose values for $\gamma$, $\alpha$, $\beta$ and $M\_0$ for different kinds of measurements and galaxies. For the late-type galaxies (which are the ones that you want to study), these parameters are shown in table 2 with values that have a certain range.

* $\alpha$ between $0.13$ and $0.17$
* $\beta$ between $0.91$ and $1.00$
* $\gamma$ between $0.08$ and $0.18$
* $M\_0$ between $19\cdot 10^{10}$ and $48\cdot 10^{10}$ solar masses

To make an approximate guess let's take the average values they present for each parameter. Then you will have the relation:

$R = 0.12\cdot M^{0.15}\cdot(1+3.8\cdot 10^{-12}M)^{0.81}$

Again with $R$ in $kpc$ and $M$ in solar masses.

With everything into account we are ready to plug-in some numbers. For example, the Milky Way galaxy is known to have a mass of $M = 10^{12}$ solar masses. This means that according to our model the Milky Way disk has a radius of $R = 26.97 \; kpc$. Well it turns out that we think the disk to extend between $170,000$ and $200,000$ light years in diameter, which translates to $26 \; kpc$ to $31\; kpc$, so we proved that the model works at least for our galaxy, even if our prediction is in the lower bound of the range.

---

Now, the devil is in the details; $R$ is not actually the radius of the disk as you might think. The radius is an ambiguous term here since disk galaxies don't have a sharp edge where stars stop to exist. The $R$ in the formula is actually the so-called ["effective radius"](https://en.wikipedia.org/wiki/Effective_radius), wich is in fact just the radius which contains half of the light emmited from the galaxy (this is an unambiguous way to measure the size of the galaxy). So, when talking about the radius $R$ I would take this into consideration, and use maybe a $1.2R$ estimate (20% larger) as my "true" radius for the galaxy.

Another important thing to consider is that the mass of the milky way has been measured including the mass of the dark matter and the mass of the gass. If you only want stellar mass, then the calculation will yield a galaxy that might be smaller.

These two are unimportant points if you are making just a highschool project and they don'tchange drastically your models either way, but I have to warn you about a consideration here. A galaxy that has a mass of $x$ solar masses is not a galaxy that has $x$ stars (planetary systems). That would certainly happen if all the stars where like the Sun, but as it turns out the vast majority of stars are less massive than the Sun, thus an $x$ solar mass galaxy will have much more than $x$ stars. If you are going to play with the Drake equiation that might be important. Finally if you search for the Milky Way data you will see that we think it has a mass of $M = 10^{12}$ solar masses, as I said, and we think it has around $2\cdot 10^{11}$ stars, so what's happening here? Shouldn't a galaxy with mass $x$ solar masses have more than $x$ stars? The problem here is, as I said previously, that we are considering the mass of the entire galaxy including gas, dust and dark matter, but we are only concerned about stellar mass.

---

I'm very interested in your project. Would you mind to tell me more about it? I'm just curious.

> **Comments:**
> **username_0**: thank you very much. In the research project, we will try to estimate the amount of communication between interstellar empires. First off we will make a simple stellar map of the galaxy. then randomly create alien civilizations in the habitable zone of the galaxy with a gaussian distribution (too far inside means too much chaos, too far outside means too little energy) and finally we use a sorting algorithm to figure out the distances between the different civilizations. we currently assume that civilizations within 20 light years of each other will be able to communicate  
> **username_1**: @A.Soreq That is awesome! I've been playing with similar simulations. The results are mindbending, for civilizations to contact each other you have to assume the galaxy has many many of them. The distances are crazy. I would love to hear more about this project, maybe you could PM me somehow when you have done it?  
> **username_0**: of course. I'd love to.  
> **username_0**: https://docs.google.com/document/d/1YZuBhxe9sIYj4Biu1fLAhB1bXaZ6qoPX87GPGcSON0g/edit?usp=sharing. as promised.  
> **username_1**: @A.Soreq That is awesome! Thank you so much. And thank you for the reference to AstronomyStack also!  

---

