<p>This is a rather complex question, for several reasons.</p>
<ul>
<li>First, galaxies come in many variations, regarding such diverse
properties as mass, morphology, and environment.</li>
<li>Second, different observational techniques, and different models,
yield different observables — you can observe the same field of the
sky with two different instruments and deduce a different
distribution of galaxies and their properties.</li>
<li>Third, as you mention, galaxies evolve, and you will not necessarily
obtain the same ratio between, say, gas mass and stellar mass, even for a
given galaxy type, at different redshifts.</li>
</ul>
<p>Nevertheless, some things can be said to be true &quot;in general&quot;:</p>
<h3>Stellar mass – gas mass relation</h3>
<p>The more massive a galaxy is (in terms of stellar mass, <span class="math-container">$M_*$</span>), the more efficient it is at forming stars. Hence, the gas <em>fraction</em> <span class="math-container">$f_\mathrm{gas} \equiv M_\mathrm{gas}/(M_\mathrm{gas}+M_*)$</span> decreases with <span class="math-container">$M_*$</span>. Moreover, although part of the gas in stars is returned to the interstellar medium (ISM), as time goes, and a galaxy forms stars, it will &quot;deplete&quot; the ISM, further reducing <span class="math-container">$f_\mathrm{gas}$</span>.</p>
<p>This can be seen in this plot from <a href="https://ui.adsabs.harvard.edu/abs/2012ApJ...760....6M/abstract" rel="nofollow noreferrer">Magdis et al. (2012)</a>, showing the gas fraction as a function of stellar mass today (open circles) and ~10 billion years ago (closed circles):</p>
<p><a href="https://i.sstatic.net/UolsW.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/UolsW.png" alt="fg" /></a></p>
<p>The galaxies used in this survey are &quot;main sequence&quot; galaxies, and other selection criteria apply as well.</p>
<h3>Stellar mass – halo mass relation</h3>
<p>The dark matter (DM) component of a galaxy is much more extended and diffuse than the baryons (because DM is collisionless), rather lying in a large &quot;halo&quot; around the gas and stars. Of course, we cannot see DM, making measurements of its mass difficult. Only in numerical simulations, we know exactly its mass.</p>
<p>The larger the DM halo mass (<span class="math-container">$M_\mathrm{h}$</span>), the more stars the galaxy has. But the relation is not straightforward. In general, <span class="math-container">$M_*$</span> increases with <span class="math-container">$M_\mathrm{h}$</span> more rapidly for low-mass galaxies, while for <span class="math-container">$M_\mathrm{h} \gtrsim 10^{12}\,M_\odot$</span> (roughly Milky Way-sized galaxies) the relation flattens out:</p>
<p>This is seen in the left panel of this plot from <a href="https://ui.adsabs.harvard.edu/abs/2013ApJ...770...57B/abstract" rel="nofollow noreferrer">Behroozi et al. (2013)</a>:</p>
<p><a href="https://i.sstatic.net/FAzrx.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/FAzrx.png" alt="MsMh" /></a></p>
<p>The different colors correspond to different epochs in the Universe. The data is from a cosmological simulation, but the simulation was calibrated to match various observations.</p>
<p>Another way to show this relation is seen in the right panel, where the stellar fraction <span class="math-container">$M_*/M_\mathrm{h}$</span> is seen to rise until around <span class="math-container">$M_\mathrm{h} \sim 10^{12}\,M_\odot$</span>, after which it decreases again.</p>
<p>Why is this? In general it is thought that star formation is suppressed at low masses because gas is more easily expelled from a shallow gravitational potential, while a high masses, <a href="https://en.wikipedia.org/wiki/Active_galactic_nucleus" rel="nofollow noreferrer">active galactic nuclei</a> become very efficient at blowing out gas, thus quenching star formation.</p>
<h3>How are masses measured?</h3>
<p>There are several techniques to measuring these masses.</p>
<p>Stellar masses are measured using known relations between the amount of stars and the amount of light from some physical process — either a single emission line, or a broader band of light. For highly star-forming galaxies, where there are still many hot O and B stars that ionize the surrounding gas, nebular lines such as H<span class="math-container">$\alpha$</span> or Ly<span class="math-container">$\alpha$</span> can be used, while for non-star-forming galaxies you can use e.g. the continuum radiation from heated dust.</p>
<p>The conversion depends on the assumed <a href="https://en.wikipedia.org/wiki/Initial_mass_function" rel="nofollow noreferrer">initial mass function</a> of the stellar population.</p>
<p>Likewise, gas masses and molecular masses may be measured knowing how much light a given amount of gas emits (at a given temperature, pressure, …).</p>
<p>Measurements of halo masses are typically done looking at the <em>width</em> of various spectral lines, thus deducing the velocity dispersion <span class="math-container">$\sigma_V$</span> of the gas and stars. Then, the total mass <span class="math-container">$M$</span> can be calculated from
<span class="math-container">$$
\sigma^2 = \frac{GM}{CR},
$$</span>
where <span class="math-container">$G$</span> is the gravitational constant, <span class="math-container">$R$</span> is the radius, and <span class="math-container">$C$</span> is a geometrical factor (see <a href="https://astronomy.stackexchange.com/a/18949/5264">this answer</a> for an explanation).</p>

CC BY-SA 4.0
2020-10-28T06:29:30.197Z
In the plot for stellar mass - gas mass relation f_gas and M_gas/M_* have been shown to have different proportionality with M_* but in the paragraph above the plot it is mentioned that both are equivalent. Can you please clarify this?
Dhruv Deshmukh
CC BY-SA 4.0
2020-10-28T06:53:51.187Z
Just want to know if my interpretation is correct. In the plot of M_h vs stellar mass M_* consider a galaxy with M_h say around 10^12, so now during the  different epochs in the universe the stellar mass of such a galaxy varied between 10^10 and 10^11.  Hence this should be the possibile range of stellar mass for the given halo mass of the galaxy under consideration right?
Dhruv Deshmukh
CC BY-SA 4.0
2020-10-28T07:02:34.200Z
Is the following correct when the galaxy is young then it should have no main sequence stars so the stellar mass must very low while there is a lot of gas so gas mass must be high. Now over time as galaxy develops the stellar mass will increase and gas mass will decrease. So will there be a point where all gas is depleted and only stellar mass is left, in such a scenario what will happen next? Also is this conversion from stellar mass to gas mass direct or some mass-energy conversion is involved? How do the stellar mass, gas mass and halo mass vary with time for a galaxy?
Dhruv Deshmukh
CC BY-SA 4.0
2020-10-28T10:08:05.420Z
@DhruvDeshmukh Sorry, I made a typo: fg is _not_ Mg/M*, but Mg / (Mg+M*), hence the different slope.
pela
CC BY-SA 4.0
2020-10-28T10:09:07.693Z
Wrt. your second question, yes, a galaxy with Mh ~ 1e12 would have M* ~ 1e10 at early epochs, and M* ~ 1e11 at late epochs (error bars denote 68% intervals, so ~1/3 fall outside this range). But most galaxies grow over time, so it's not necessarily the same galaxy.
pela
CC BY-SA 4.0
2020-10-28T10:12:50.617Z
Wrt. question #3, stars reach the MS rather fast, compared to the time scale of galaxy formation. It's true that, as galaxies evolve, they use more and more gas, but firstly some of it is returned, and secondly they keep accreting gas from the surroundings. Some galaxies are gas-depleted, but this is more likely due to earlier mergers where some gas is expelled from the galaxies, and the rest is used in one or more starbursts, quenching its star formation. These are the massive ellipticals. Spirals, on the other hand, have ongoing SF, and won't deplete until many billion years have passed.
pela
CC BY-SA 4.0
2020-10-28T10:25:14.790Z
You ask "what will happen next": So, for these quenched galaxies, the higher-mass stars eventually die out, while the lower-mass stars hang around for longer. Since low-mass stars are red/orange, this is the color of such galaxies. The lowest-mass star live for 100 times the current age of the Universe or so. I'm not sure I understand with "_is this conversion from stellar mass to gas mass direct or some mass-energy conversion is involved_". If you mean whether some mass is "lost" to energy, then yes, a star converts some of its mass to energy, but this is a minor fraction.
pela
CC BY-SA 4.0
2020-10-28T10:27:54.000Z
You final question is also quite complex, and should perhaps be posted as a second question. I see now that you asked it in your original post, and I can try and find some references, but will have to get back to work for now :)
pela
CC BY-SA 4.0
2020-10-28T10:47:34.677Z
@pela "Some galaxies are gas-depleted, but this is more likely due to earlier mergers" -- you're ignoring some other processes, like ram-pressure and tidal stripping, and possibly strong SF and AGN feedback, that can remove gas from a galaxy.
Peter Erwin
CC BY-SA 4.0
2020-10-28T16:44:36.513Z
@pela One more small question. You mentioned in your comment that stars convert a small fraction of their mass to energy, which essentially means that their mass is almost constant. Does this apply only when star is in main sequence or is it always true?
Dhruv Deshmukh
CC BY-SA 4.0
2020-10-29T09:14:23.900Z
@PeterErwin Yes, good point!
pela
CC BY-SA 4.0
2020-10-29T09:19:04.023Z
@DhruvDeshmukh Not exactly: They also lose mass due to stellar winds. I didn't consider it "lost", because the atoms still exist, they're just ejected into the surrounding interstellar medium (in contrast, the mass loss dut to energy production is converted to photons). Massive stars with extreme luminosities can lose a quite significant fraction of their mass during the main sequence. Smaller stars don't lose much at the MS, I think, but later when they swell to become red giants, their outer layers become more loosely bound and is hence more easily ejected.
pela
CC BY-SA 4.0
2020-10-31T17:24:55.310Z
@pela You said that the confidence interval is 68% in the stellar mass-halo mass relation plot. Could you please tell me the sigma/√n so that I can calculate the possible range? Also why does the distribution narrow down at 10^12 and widen as we go left and right in the plot?
Dhruv Deshmukh
CC BY-SA 4.0
2020-10-31T21:36:26.143Z
@DhruvDeshmukh According to their description, the lines show the mean values for central galaxies (i.e. not satellites), and the error bars include both systematic and statistical uncertainties, calculated for a fixed cosmological model. You can read more about their model, and the number of galaxies that they used for their statistics, in Behroozi+ 13. I don't know the reason for the widening of the error bars, but one effect could be small-number statistics for large galaxies, and the larger observational uncertainties for faint galaxies.
pela
CC BY-SA 4.0
2020-11-02T09:57:05.870Z
@pela I have added my thinking about the change in these masses of a galaxy over time. If possible please go through it and correct me if I am wrong some where.
Dhruv Deshmukh
CC BY-SA 4.0
2020-11-02T13:14:53.213Z
@DhruvDeshmukh Your description of galaxy formation is similar to the so-called "monolithic collapse", or top-down scenario, described by [Eggen, Lynden-Bell, & Sandage (1962)], but this scenario isn't really thought to be true anymore. More probably galaxies form in a "bottom-up" fashion, where mini-halos of M ~ 1e5 Mo or so are the first structures to form, later merging to form larger structures. But in general your description seems to be okay, although I don't think the evolution with time is nearly as strong as the dependence on halo mass.
pela
CC BY-SA 4.0
2020-11-02T13:16:14.460Z
[This question](https://astronomy.stackexchange.com/q/39632/5264) seems to be what you're after. I don't think the current answer is what you're looking for, but I don't have the time at the moment to write a good one myself (sorry).
pela
CC BY-SA 4.0
2020-10-26T16:08:12.687Z
2020-10-28T10:14:06.717Z
2020-10-28T10:14:06.717Z
pela
<p>I am doing an introductory course in astrophysics and astronomy undergrad level.
In it we were taught that the main components of matter in a galaxy are dark matter halo (<span class="math-container">$M_h$</span>), gas (<span class="math-container">$M_{gas}$</span>) and stars (<span class="math-container">$M_*$</span>). Thus the mass of the galaxy can be given as the sum of these three components. I want to read more about this but when I search this on net I don't get any appropriate results or they are very advanced. I guess this is because the actual terminology and nomenclature of these masses is different.
Thus if anyone can give me accurate terminology of these masses it would be helpful.</p>
<p>Additionally we were told that given the mass of dark matter halo the other two masses can have only a certain range of values. I want to know how we can determine these range of values possible for a galaxy.
Also how do these values change as the galaxy ages?
Any links to resources regarding the above topics are also welcome.</p>
<p>Edit 1: Regarding my question above about the change in mass of galaxies over time I am giving my thinking. Please correct me if I am wrong somewhere.</p>
<p>Initially galaxy consists of only dark matter and gas. This initially contracts but later undergoes segmentation to form stars. Thus initially there should be decrease in <span class="math-container">$M_{gas}$</span>. But slowly some gas is returned as stars die and also gathered from surroundings, hence the gas mass should remain constant or decrease at a very slow rate. Now for the stars initially <span class="math-container">$M_*$</span> should increase due to creation of new stars. But after some time as some of the early stars start dying out there will be simultaneous birth and death of stars, hence the <span class="math-container">$M_*$</span> should fairly remain constant during this period. Finally as the star formation rate decreases due to depletion of <span class="math-container">$M_{gas}$</span> the birth of new stars will be reduced. Slowly more and more stars will start dying and <span class="math-container">$M_*$</span> will be depleted leaving behind only heavier elements. Hence <span class="math-container">$M_*$</span> should decrease during this period. Finally coming to halo mass, I am still not sure about it. I think it should not change much over the lifetime of the galaxy because dark matter does not seem to interact with the baryonic mass. I know that these changes will be slow but can anyone give me a rough number on the slowness of these changes?</p>

<galaxy><dark-matter><galactic-halo>
The contribution of components like dark matter, stars and gas to the mass of a galaxy
Dhruv Deshmukh
# The contribution of components like dark matter, stars and gas to the mass of a galaxy

(Asked by: username_0 on 2020-10-26)

I am doing an introductory course in astrophysics and astronomy undergrad level.
In it we were taught that the main components of matter in a galaxy are dark matter halo ($M\_h$), gas ($M\_{gas}$) and stars ($M\_\*$). Thus the mass of the galaxy can be given as the sum of these three components. I want to read more about this but when I search this on net I don't get any appropriate results or they are very advanced. I guess this is because the actual terminology and nomenclature of these masses is different.
Thus if anyone can give me accurate terminology of these masses it would be helpful.

Additionally we were told that given the mass of dark matter halo the other two masses can have only a certain range of values. I want to know how we can determine these range of values possible for a galaxy.
Also how do these values change as the galaxy ages?
Any links to resources regarding the above topics are also welcome.

Edit 1: Regarding my question above about the change in mass of galaxies over time I am giving my thinking. Please correct me if I am wrong somewhere.

Initially galaxy consists of only dark matter and gas. This initially contracts but later undergoes segmentation to form stars. Thus initially there should be decrease in $M\_{gas}$. But slowly some gas is returned as stars die and also gathered from surroundings, hence the gas mass should remain constant or decrease at a very slow rate. Now for the stars initially $M\_\*$ should increase due to creation of new stars. But after some time as some of the early stars start dying out there will be simultaneous birth and death of stars, hence the $M\_\*$ should fairly remain constant during this period. Finally as the star formation rate decreases due to depletion of $M\_{gas}$ the birth of new stars will be reduced. Slowly more and more stars will start dying and $M\_\*$ will be depleted leaving behind only heavier elements. Hence $M\_\*$ should decrease during this period. Finally coming to halo mass, I am still not sure about it. I think it should not change much over the lifetime of the galaxy because dark matter does not seem to interact with the baryonic mass. I know that these changes will be slow but can anyone give me a rough number on the slowness of these changes?

**Tags:** galaxy, dark-matter, galactic-halo


---

## 1 answers

### Answer by username_1 on 2020-10-26. Score: 6 | Accepted answer

This is a rather complex question, for several reasons.

* First, galaxies come in many variations, regarding such diverse
  properties as mass, morphology, and environment.
* Second, different observational techniques, and different models,
  yield different observables — you can observe the same field of the
  sky with two different instruments and deduce a different
  distribution of galaxies and their properties.
* Third, as you mention, galaxies evolve, and you will not necessarily
  obtain the same ratio between, say, gas mass and stellar mass, even for a
  given galaxy type, at different redshifts.

Nevertheless, some things can be said to be true "in general":

### Stellar mass – gas mass relation

The more massive a galaxy is (in terms of stellar mass, $M\_\*$), the more efficient it is at forming stars. Hence, the gas *fraction* $f\_\mathrm{gas} \equiv M\_\mathrm{gas}/(M\_\mathrm{gas}+M\_\*)$ decreases with $M\_\*$. Moreover, although part of the gas in stars is returned to the interstellar medium (ISM), as time goes, and a galaxy forms stars, it will "deplete" the ISM, further reducing $f\_\mathrm{gas}$.

This can be seen in this plot from [Magdis et al. (2012)](https://ui.adsabs.harvard.edu/abs/2012ApJ...760....6M/abstract), showing the gas fraction as a function of stellar mass today (open circles) and ~10 billion years ago (closed circles):

[![fg](https://i.sstatic.net/UolsW.png)](https://i.sstatic.net/UolsW.png)

The galaxies used in this survey are "main sequence" galaxies, and other selection criteria apply as well.

### Stellar mass – halo mass relation

The dark matter (DM) component of a galaxy is much more extended and diffuse than the baryons (because DM is collisionless), rather lying in a large "halo" around the gas and stars. Of course, we cannot see DM, making measurements of its mass difficult. Only in numerical simulations, we know exactly its mass.

The larger the DM halo mass ($M\_\mathrm{h}$), the more stars the galaxy has. But the relation is not straightforward. In general, $M\_\*$ increases with $M\_\mathrm{h}$ more rapidly for low-mass galaxies, while for $M\_\mathrm{h} \gtrsim 10^{12}\,M\_\odot$ (roughly Milky Way-sized galaxies) the relation flattens out:

This is seen in the left panel of this plot from [Behroozi et al. (2013)](https://ui.adsabs.harvard.edu/abs/2013ApJ...770...57B/abstract):

[![MsMh](https://i.sstatic.net/FAzrx.png)](https://i.sstatic.net/FAzrx.png)

The different colors correspond to different epochs in the Universe. The data is from a cosmological simulation, but the simulation was calibrated to match various observations.

Another way to show this relation is seen in the right panel, where the stellar fraction $M\_\*/M\_\mathrm{h}$ is seen to rise until around $M\_\mathrm{h} \sim 10^{12}\,M\_\odot$, after which it decreases again.

Why is this? In general it is thought that star formation is suppressed at low masses because gas is more easily expelled from a shallow gravitational potential, while a high masses, [active galactic nuclei](https://en.wikipedia.org/wiki/Active_galactic_nucleus) become very efficient at blowing out gas, thus quenching star formation.

### How are masses measured?

There are several techniques to measuring these masses.

Stellar masses are measured using known relations between the amount of stars and the amount of light from some physical process — either a single emission line, or a broader band of light. For highly star-forming galaxies, where there are still many hot O and B stars that ionize the surrounding gas, nebular lines such as H$\alpha$ or Ly$\alpha$ can be used, while for non-star-forming galaxies you can use e.g. the continuum radiation from heated dust.

The conversion depends on the assumed [initial mass function](https://en.wikipedia.org/wiki/Initial_mass_function) of the stellar population.

Likewise, gas masses and molecular masses may be measured knowing how much light a given amount of gas emits (at a given temperature, pressure, …).

Measurements of halo masses are typically done looking at the *width* of various spectral lines, thus deducing the velocity dispersion $\sigma\_V$ of the gas and stars. Then, the total mass $M$ can be calculated from
$$
\sigma^2 = \frac{GM}{CR},
$$
where $G$ is the gravitational constant, $R$ is the radius, and $C$ is a geometrical factor (see [this answer](https://astronomy.stackexchange.com/a/18949/5264) for an explanation).

> **Comments:**
> **username_0**: In the plot for stellar mass - gas mass relation f_gas and M_gas/M_* have been shown to have different proportionality with M_* but in the paragraph above the plot it is mentioned that both are equivalent. Can you please clarify this?  
> **username_0**: Just want to know if my interpretation is correct. In the plot of M_h vs stellar mass M_* consider a galaxy with M_h say around 10^12, so now during the  different epochs in the universe the stellar mass of such a galaxy varied between 10^10 and 10^11.  Hence this should be the possibile range of stellar mass for the given halo mass of the galaxy under consideration right?  
> **username_0**: Is the following correct when the galaxy is young then it should have no main sequence stars so the stellar mass must very low while there is a lot of gas so gas mass must be high. Now over time as galaxy develops the stellar mass will increase and gas mass will decrease. So will there be a point where all gas is depleted and only stellar mass is left, in such a scenario what will happen next? Also is this conversion from stellar mass to gas mass direct or some mass-energy conversion is involved? How do the stellar mass, gas mass and halo mass vary with time for a galaxy?  
> **username_1**: @DhruvDeshmukh Sorry, I made a typo: fg is _not_ Mg/M*, but Mg / (Mg+M*), hence the different slope.  
> **username_1**: Wrt. your second question, yes, a galaxy with Mh ~ 1e12 would have M* ~ 1e10 at early epochs, and M* ~ 1e11 at late epochs (error bars denote 68% intervals, so ~1/3 fall outside this range). But most galaxies grow over time, so it's not necessarily the same galaxy.  
> **username_1**: Wrt. question #3, stars reach the MS rather fast, compared to the time scale of galaxy formation. It's true that, as galaxies evolve, they use more and more gas, but firstly some of it is returned, and secondly they keep accreting gas from the surroundings. Some galaxies are gas-depleted, but this is more likely due to earlier mergers where some gas is expelled from the galaxies, and the rest is used in one or more starbursts, quenching its star formation. These are the massive ellipticals. Spirals, on the other hand, have ongoing SF, and won't deplete until many billion years have passed.  
> **username_1**: You ask "what will happen next": So, for these quenched galaxies, the higher-mass stars eventually die out, while the lower-mass stars hang around for longer. Since low-mass stars are red/orange, this is the color of such galaxies. The lowest-mass star live for 100 times the current age of the Universe or so. I'm not sure I understand with "_is this conversion from stellar mass to gas mass direct or some mass-energy conversion is involved_". If you mean whether some mass is "lost" to energy, then yes, a star converts some of its mass to energy, but this is a minor fraction.  
> **username_1**: You final question is also quite complex, and should perhaps be posted as a second question. I see now that you asked it in your original post, and I can try and find some references, but will have to get back to work for now :)  
> **username_2**: @username_1 "Some galaxies are gas-depleted, but this is more likely due to earlier mergers" -- you're ignoring some other processes, like ram-pressure and tidal stripping, and possibly strong SF and AGN feedback, that can remove gas from a galaxy.  
> **username_0**: @username_1 One more small question. You mentioned in your comment that stars convert a small fraction of their mass to energy, which essentially means that their mass is almost constant. Does this apply only when star is in main sequence or is it always true?  
> **username_1**: @PeterErwin Yes, good point!  
> **username_1**: @DhruvDeshmukh Not exactly: They also lose mass due to stellar winds. I didn't consider it "lost", because the atoms still exist, they're just ejected into the surrounding interstellar medium (in contrast, the mass loss dut to energy production is converted to photons). Massive stars with extreme luminosities can lose a quite significant fraction of their mass during the main sequence. Smaller stars don't lose much at the MS, I think, but later when they swell to become red giants, their outer layers become more loosely bound and is hence more easily ejected.  
> **username_0**: @username_1 You said that the confidence interval is 68% in the stellar mass-halo mass relation plot. Could you please tell me the sigma/√n so that I can calculate the possible range? Also why does the distribution narrow down at 10^12 and widen as we go left and right in the plot?  
> **username_1**: @DhruvDeshmukh According to their description, the lines show the mean values for central galaxies (i.e. not satellites), and the error bars include both systematic and statistical uncertainties, calculated for a fixed cosmological model. You can read more about their model, and the number of galaxies that they used for their statistics, in Behroozi+ 13. I don't know the reason for the widening of the error bars, but one effect could be small-number statistics for large galaxies, and the larger observational uncertainties for faint galaxies.  
> **username_0**: @username_1 I have added my thinking about the change in these masses of a galaxy over time. If possible please go through it and correct me if I am wrong some where.  
> **username_1**: @DhruvDeshmukh Your description of galaxy formation is similar to the so-called "monolithic collapse", or top-down scenario, described by [Eggen, Lynden-Bell, & Sandage (1962)], but this scenario isn't really thought to be true anymore. More probably galaxies form in a "bottom-up" fashion, where mini-halos of M ~ 1e5 Mo or so are the first structures to form, later merging to form larger structures. But in general your description seems to be okay, although I don't think the evolution with time is nearly as strong as the dependence on halo mass.  
> **username_1**: [This question](https://astronomy.stackexchange.com/q/39632/5264) seems to be what you're after. I don't think the current answer is what you're looking for, but I don't have the time at the moment to write a good one myself (sorry).  

---

