<p>The luminosity of Trappist-1 is estimated to be $5.25\times 10^{-4}\ L_{\odot}$, but it has not always been like this.</p>

<p>The luminosity of a brown dwarf decreases with time and it this measured luminosity (along with the spectral type) that allows an <em>estimate</em> of the mass and a <em>lower limit</em> to the age using stellar evolutionary models.</p>

<p>If I look at the <a href="https://arxiv.org/abs/1503.04107" rel="nofollow noreferrer">Baraffe et al. (2015)</a> low-mass evolutionary models and look at the locus of luminosity versus time for a $0.08\ M_{\odot}$ star like Trappist-1, you can see that the current luminosity implies an age of $\sim 500$ million years. But if you go back in time, the star was more luminous and for that reason, planets that are <em>currently</em> in the habitable zone (said to be planets e,f,g) were not so in the past.</p>

<p>The details of a habitable zone (HZ) calculation can be complex, but basically the radius of the habitable zone scales as the square root of the luminosity. If planets d and h are <em>not</em> currently in the HZ then we can use these as a conservative definition of the HZ boundary. </p>

<p>From this (and using the published orbital radii of the planets), I can see that if the luminosity is increased by a factor of 9, then <em>none</em> of the planets b-g are in the HZ, it is larger than all of their orbits. Trappist-1 had a luminosity that was 9 times larger when it was younger than 27 million years old. On the other hand, if I want to move the HZ just outside the orbit of planet e (and simultaneously include planet h inside the HZ), then this would occur when Trappist-1 was at an age of 206 million years. As a final thought, you can see from this particular model that Trappist-1 may fade by a further factor of two as it gets older. This <em>decreases</em> the HZ radius by a factor of 1.41 and would mean that g (and possibly f) would fall outside the HZ, whilst d (and possibly c) would be brought into the HZ.</p>

<p>It should be noted however that: different models give slightly different results, these loci are mass dependent and the mass is not known, it is inferred from the same models using a temperature estimate (which is also uncertain). So whilst my qualitative conclusions about the <em>past</em> location of the HZ are likely to be correct (though the detailed age numbers are model-dependent), the future behaviour of the HZ is more uncertain because Trappist-1 may be slightly more massive than assumed and <em>already</em> reached its minimum luminosity.</p>

<p><a href="https://i.sstatic.net/g9a7q.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/g9a7q.png" alt="Luminosity evolution of Trappist-1"></a>
<em>The luminosity evolution of Trappist-1, assuming a mass of $0.08M_{\odot}$ and the models of Baraffe et al. (2015). The horizontal dashed line marks the best estimate of its current luminosity, for which planets e-g are said to be in the HZ. If we go back in time, the rightmost vertical dashed line marks the age below which the luminosity increases to the point that e becomes too hot to be habitable. Then further back, the leftmost vertical dashed line marks the point where all the currently known planets (b-h) become uninhabitable.</em></p>

<p>So the answer to your question is quite uncertain and depends critically on the age of Trappist-1 now and of course <strong>whether the planets were always at the orbital radii they are now</strong>$^{*}$. As you can see from the plot above (note the logarithmic scale on the x-axis), the luminosity evolution referred to above takes place early-on. If Trappist-1 could be as young as 500 million years, then life on planet e may only have been possible for 300 million years. However, if the star is slightly more massive and is 10 billion years old, then life has had 9.8 billion years to get going.</p>

<p>If you are talking about planet f, then it has had slightly longer ($\sim +100$ million years) inside the HZ, and planet g slightly longer ($\sim +70$ million years) again. Planet h will have spent comparatively little time (in the past), less than a few hundred million years, inside the HZ.</p>

<p>$^{*}$ The abstract of the discovery paper by <a href="http://www.eso.org/public/archives/releases/sciencepapers/eso1706/eso1706a.pdf" rel="nofollow noreferrer">Gillon et al. (2017)</a> briefly discusses the possibility that the planets migrated <em>inwards</em> after formation, through a process of "disc-driven migration". If so, then this will <em>not</em> alter the discussion above. Disc around very low-mass stars may be longer-lived than those around higher mass stars, but have essentially dispersed after $\sim 10-20$ million years (<a href="http://iopscience.iop.org/article/10.1088/0004-637X/695/2/1210/meta" rel="nofollow noreferrer">Kennedy &amp; Kenyon 2009</a>; <a href="http://adsabs.harvard.edu/abs/2013MNRAS.429..903D" rel="nofollow noreferrer">Dawson et al. 2013</a>; <a href="https://arxiv.org/abs/1611.07416" rel="nofollow noreferrer">Binks &amp; Jeffries 2017</a>), and the planetary configuration would have to be settled where it is now by the time the disc has gone.</p>

CC BY-SA 3.0
2017-02-25T14:49:32.140Z
If I understand correctly, the article posited that the planets initially formed further out; past the frost line for the star at that time.  They then migrated inward.  That makes determining when any of the planets was in the HZ for that time notably more difficult.  Though I have not familiarized myself with their article and results enough to know if their models include specific enough details on the potential past evolution of the system to let one make reasonable estimates (within their model(s)).
zibadawa timmy
CC BY-SA 3.0
2017-02-25T15:09:13.173Z
@zibadawatimmy The posited mechanism is disc-driven inward migration. If so, then as disc lifetimes around low-mass stars are <10 Myr, then it wouldn't affect any of the conclusions.
ProfRob
CC BY-SA 3.0
2017-02-25T10:50:04.880Z
2017-02-25T22:40:12.673Z
2017-02-25T22:40:12.673Z
ProfRob
<p>The planets orbiting TRAPPIST-1 are in orbits around their star much closer than Earth orbits Sol. However, since TRAPPIST-1 is a cool brown dwarf, some of the planets are in an "inhabitable zone", where the temperature would not be too extreme for life as we know it.</p>

<p>What I wonder is, how long have those planets been in an inhabitable zone? The <a href="https://en.wikipedia.org/wiki/TRAPPIST-1" rel="noreferrer">wikipedia page</a> says that TRAPPIST-1 is older than 1 Gyr. But, how much of that time has it been a cool brown dwarf?</p>

<p>In other words, since the abiogenesis of life and the evolution of living organisms has dependency on time, how much time has life had a chance to develop in the star's habitable zone?</p>

<p>For this question, I'm not concerned about the chemistry or other characteristics of the planets themselves.</p>

CC BY-SA 3.0
2017-02-24T21:39:36.920Z
TRAPPIST-1 formed as a cool brown dwarf, and has been so its entire life.  That being said, the planets haven't always been at the orbits which they currently occupy.
Phiteros
CC BY-SA 3.0
2017-02-24T22:34:03.317Z
TRAPPIST-1 is an M-type red dwarf. It is not a brown dwarf; the term "ultra-cool dwarf" spans both M and L class objects and is therefore not limited to either.
HDE 226868
CC BY-SA 3.0
2017-02-25T10:51:42.123Z
@Phiteros The temperature of the brown dwarf is not the main issue (though it was warmer in the past). The main issue is that the luminosity was *considerably* higher in the past, such that all the planets were closer than the habitable zone at a time in the past.
ProfRob
CC BY-SA 3.0
2017-02-25T11:00:24.470Z
@HDE226868 It is not possible to rule out that Trappist-1 is a brown dwarf. The mass is quoted as $0.080\pm 0.009 M_{\odot}$, which allows it to be a star or a brown dwarf. And I think that error bar is ridiculously optimistic.
ProfRob
CC BY-SA 3.0
2017-02-25T13:28:34.523Z
@RobJeffries I'm sorry; I automatically assumed it wasn't. I hadn't seen anything suggesting it was a brown dwarf.
HDE 226868
<exoplanet><trappist-1>
How long have the TRAPPIST-1 planets been in a habitable zone?
user151841
# How long have the TRAPPIST-1 planets been in a habitable zone?

(Asked by: username_0 on 2017-02-24)

The planets orbiting TRAPPIST-1 are in orbits around their star much closer than Earth orbits Sol. However, since TRAPPIST-1 is a cool brown dwarf, some of the planets are in an "inhabitable zone", where the temperature would not be too extreme for life as we know it.

What I wonder is, how long have those planets been in an inhabitable zone? The [wikipedia page](https://en.wikipedia.org/wiki/TRAPPIST-1) says that TRAPPIST-1 is older than 1 Gyr. But, how much of that time has it been a cool brown dwarf?

In other words, since the abiogenesis of life and the evolution of living organisms has dependency on time, how much time has life had a chance to develop in the star's habitable zone?

For this question, I'm not concerned about the chemistry or other characteristics of the planets themselves.

**Tags:** exoplanet, trappist-1

> **Comments:**
> **username_1**: TRAPPIST-1 formed as a cool brown dwarf, and has been so its entire life.  That being said, the planets haven't always been at the orbits which they currently occupy.  
> **username_2**: TRAPPIST-1 is an M-type red dwarf. It is not a brown dwarf; the term "ultra-cool dwarf" spans both M and L class objects and is therefore not limited to either.  
> **username_3**: @username_1 The temperature of the brown dwarf is not the main issue (though it was warmer in the past). The main issue is that the luminosity was *considerably* higher in the past, such that all the planets were closer than the habitable zone at a time in the past.  
> **username_3**: @username_2 It is not possible to rule out that Trappist-1 is a brown dwarf. The mass is quoted as $0.080\pm 0.009 M_{\odot}$, which allows it to be a star or a brown dwarf. And I think that error bar is ridiculously optimistic.  
> **username_2**: @RobJeffries I'm sorry; I automatically assumed it wasn't. I hadn't seen anything suggesting it was a brown dwarf.  

---

## 1 answers

### Answer by username_3 on 2017-02-25. Score: 8 | Accepted answer

The luminosity of Trappist-1 is estimated to be $5.25\times 10^{-4}\ L\_{\odot}$, but it has not always been like this.

The luminosity of a brown dwarf decreases with time and it this measured luminosity (along with the spectral type) that allows an *estimate* of the mass and a *lower limit* to the age using stellar evolutionary models.

If I look at the [Baraffe et al. (2015)](https://arxiv.org/abs/1503.04107) low-mass evolutionary models and look at the locus of luminosity versus time for a $0.08\ M\_{\odot}$ star like Trappist-1, you can see that the current luminosity implies an age of $\sim 500$ million years. But if you go back in time, the star was more luminous and for that reason, planets that are *currently* in the habitable zone (said to be planets e,f,g) were not so in the past.

The details of a habitable zone (HZ) calculation can be complex, but basically the radius of the habitable zone scales as the square root of the luminosity. If planets d and h are *not* currently in the HZ then we can use these as a conservative definition of the HZ boundary.

From this (and using the published orbital radii of the planets), I can see that if the luminosity is increased by a factor of 9, then *none* of the planets b-g are in the HZ, it is larger than all of their orbits. Trappist-1 had a luminosity that was 9 times larger when it was younger than 27 million years old. On the other hand, if I want to move the HZ just outside the orbit of planet e (and simultaneously include planet h inside the HZ), then this would occur when Trappist-1 was at an age of 206 million years. As a final thought, you can see from this particular model that Trappist-1 may fade by a further factor of two as it gets older. This *decreases* the HZ radius by a factor of 1.41 and would mean that g (and possibly f) would fall outside the HZ, whilst d (and possibly c) would be brought into the HZ.

It should be noted however that: different models give slightly different results, these loci are mass dependent and the mass is not known, it is inferred from the same models using a temperature estimate (which is also uncertain). So whilst my qualitative conclusions about the *past* location of the HZ are likely to be correct (though the detailed age numbers are model-dependent), the future behaviour of the HZ is more uncertain because Trappist-1 may be slightly more massive than assumed and *already* reached its minimum luminosity.

[![Luminosity evolution of Trappist-1](https://i.sstatic.net/g9a7q.png)](https://i.sstatic.net/g9a7q.png)
*The luminosity evolution of Trappist-1, assuming a mass of $0.08M\_{\odot}$ and the models of Baraffe et al. (2015). The horizontal dashed line marks the best estimate of its current luminosity, for which planets e-g are said to be in the HZ. If we go back in time, the rightmost vertical dashed line marks the age below which the luminosity increases to the point that e becomes too hot to be habitable. Then further back, the leftmost vertical dashed line marks the point where all the currently known planets (b-h) become uninhabitable.*

So the answer to your question is quite uncertain and depends critically on the age of Trappist-1 now and of course **whether the planets were always at the orbital radii they are now**$^{\*}$. As you can see from the plot above (note the logarithmic scale on the x-axis), the luminosity evolution referred to above takes place early-on. If Trappist-1 could be as young as 500 million years, then life on planet e may only have been possible for 300 million years. However, if the star is slightly more massive and is 10 billion years old, then life has had 9.8 billion years to get going.

If you are talking about planet f, then it has had slightly longer ($\sim +100$ million years) inside the HZ, and planet g slightly longer ($\sim +70$ million years) again. Planet h will have spent comparatively little time (in the past), less than a few hundred million years, inside the HZ.

$^{\*}$ The abstract of the discovery paper by [Gillon et al. (2017)](http://www.eso.org/public/archives/releases/sciencepapers/eso1706/eso1706a.pdf) briefly discusses the possibility that the planets migrated *inwards* after formation, through a process of "disc-driven migration". If so, then this will *not* alter the discussion above. Disc around very low-mass stars may be longer-lived than those around higher mass stars, but have essentially dispersed after $\sim 10-20$ million years ([Kennedy & Kenyon 2009](http://iopscience.iop.org/article/10.1088/0004-637X/695/2/1210/meta); [Dawson et al. 2013](http://adsabs.harvard.edu/abs/2013MNRAS.429..903D); [Binks & Jeffries 2017](https://arxiv.org/abs/1611.07416)), and the planetary configuration would have to be settled where it is now by the time the disc has gone.

> **Comments:**
> **username_4**: If I understand correctly, the article posited that the planets initially formed further out; past the frost line for the star at that time.  They then migrated inward.  That makes determining when any of the planets was in the HZ for that time notably more difficult.  Though I have not familiarized myself with their article and results enough to know if their models include specific enough details on the potential past evolution of the system to let one make reasonable estimates (within their model(s)).  
> **username_3**: @username_4 The posited mechanism is disc-driven inward migration. If so, then as disc lifetimes around low-mass stars are <10 Myr, then it wouldn't affect any of the conclusions.  

---

