How Habitable Zones depend on Eccentricity

[Widdekind]

Please ponder a potentially Habitable Planet, complete w/ Atmosphere & Oceans sufficient to support possible Life. Now, imagine that this planet's orbit is notably Eccentric. Thus, this planet dives down in, near to its parent star, rapidly absorbing copious quantities of energy (surely, a short Summer). Then, the planet pulls back out, far from its parent star, and gradually cools (long Winter).

 

SPECULATION: The parent star's Habitable Zone (HZ) basically comprises that region, wherein that star's observed Energy Flux (Watts per area) is comparable to that of our Sun, at our Earth (~1400 W / m²).

 

But, b/c a potentially habitable planet can absorb energy, into its Oceans, we can consider an averaged Energy Flux, over one full revolution of the planet (to wit, one Orbital Year):

[math]Time\;Averaged\;Enegy\;Flux \equiv \frac{1}{Orbital Period} \int Energy Flux \; \partial{t}[/math]

But:

[math]Energy\;Flux = \frac{L_{*}}{4 \pi r^{2}}[/math]

And (?):

[math]\partial{t} = \frac{\partial{\theta}}{\frac{\partial{\theta}}{\partial{t}}} = \frac{\partial{\theta}}{\dot{\theta}}[/math]

So:

[math]\overline{\Phi} \equiv \frac{1}{P} \int_{0}^{2 \pi} \frac{L_{*}}{4 \pi r^{2}} \frac{\partial{\theta}}{\dot{\theta}}[/math]

Now, it is well-known, that Planetary Orbits conserve (Specific) Angular-Momentum:

[math]\widetilde{\ell} \equiv \widetilde{r} \times \widetilde{v} = Constant[/math]

And so we can consider (using well-known Vector Identities):

[math]\ell^{2} = \left( \widetilde{r} \times \widetilde{v} \right) \bullet \left( \widetilde{r} \times \widetilde{v} \right) = \left( \widetilde{r} \bullet \widetilde{r} \right)\left( \widetilde{v} \bullet \widetilde{v} \right) - \left(\widetilde{r}\bullet\widetilde{v}\right)\left(\widetilde{r}\bullet\widetilde{v}\right)=\left(r^{2}\right)\left(v^{2}\right)-\left(r\dot{r}\right)\left(r\dot{r}\right)[/math]

where we have decomposed the Orbital Velocity (v) (technically, the velocity of the Reduced Mass, of the Reduced One-Body Problem) as:

[math]\vec{v} = \dot{r} \; \hat{r} + r \; \dot{\theta} \; \hat{\theta}[/math]

So:

[math]v^{2} = \vec{v} \bullet \vec{v} = \dot{r}^{2} + r^{2} \; \dot{\theta}^{2}[/math]

and:

[math]\ell^{2} = \left( r^{2} \right) \left( v^{2} - \dot{r}^{2}\right) = r^{4} \; \dot{\theta}^{2} [/math]

Taking the square-root, and plugging back in above, we have that:

[math]\overline{\Phi} \equiv \frac{1}{P} \int_{0}^{2 \pi} \frac{L_{*}}{4 \pi \ell} \partial{\theta} = \frac{L_{*}}{2 P \ell} [/math]

Now, it can be shown (see below post) that:

[math]\ell = \sqrt{G M_{*} a (1-e^{2})}[/math]

and:

[math]P^{2} = a^{3} \frac{4 \pi^{2}}{G M_{*}}[/math]

Therefore, we have that:

[math]\overline{\Phi} = \frac{L_{*}}{4 \pi a^{2}} \frac{1}{\sqrt{1-e^{2}}} [/math]

 

CONCLUSION: As a potentially habitable planet's Eccentricity (e) increases, its Semi-Major Axis (a) must also increase, to keep constant its annual energy intake. This derivation assumes that the habitable planet's Global Temperature, being moderated by its Oceans, does not change considerably over the course of its orbit, and that, thusly, neither does its thermal re-radition rate. This effect is minute for most Eccentricities.

[Sayonara]

this planet's orbit is notably Eccentric. Thus, this planet dives down in, near to its parent star, rapidly absorbing copious quantities of energy (surely, a short Summer). Then, the planet pulls back out, far from its parent star, and gradually cools (long Winter).

The change in temperature in different planetary seasons has nothing to do with that planet's proximity to the sun. It is due to the tilt of the planet's axis relative to its plane of revolution.

 

I am not sure what significance you think a "Global Temperature" has or why anyone should be that interested in orbital eccentricities, because you have not really explained your reasoning. What I can say though is that your premises are faulty and therefore so is your conclusion.

[Martin]

Please ponder a potentially Habitable Planet, complete w/ Atmosphere & Oceans sufficient to support possible Life. Now, imagine that this planet's orbit is notably Eccentric. Thus, this planet dives down in, near to its parent star, rapidly absorbing copious quantities of energy (surely, a short Summer). Then, the planet pulls back out, far from its parent star, and gradually cools (long Winter).

 

...

...

CONCLUSION: As a potentially habitable planet's Eccentricity (e) increases, its Semi-Major Axis (a) must also increase, to keep constant its annual energy intake. This derivation assumes that the habitable planet's Global Temperature, being moderated by its Oceans, does not change considerably over the course of its orbit, and that, thusly, neither does its thermal re-radition rate. This effect is minute for most Eccentricities.

 

Hi Widdekind!

 

I think your conclusion is correct. I haven't checked every detail. And you refer to something you are going to supply in a further post. But it is intuitive.

You are assuming a planet with ocean where the ocean serves to stabilize and equalize temp. With that simplifying assumption what matters most is the annual flux.

 

Even so, the effect of eccentricity might be small, even for e = 0.1 or e = 0.2. Worth considering though.

 

Two atmospheric effects are also important, and work in opposite directions:

 

* reflection off cloud-tops (increasing albedo) reducing effective flux that is actually absorbed, and

* greenhouse

 

The more massive terrestrial-type planets can be expected to have retained denser atmosphere and to have more of these two effects.

 

====================

 

I realize you have a graduate degree and an intense interest in the physics of habitable planet formation. Factors affecting how many habitable planets the current searches are likely to find, and what kinds of habitable-zone planets will be most abundant.

 

I have seen a number of your short message-board "papers" at Astronomy Forum.

 

I note that the Admin at astronomy forum thanked you for sharing. I agree with him that your stuff is interesting and food for thought.

===

I also think writing it up is a learning exercise for you. I believe you are currently going thru Bradley Carroll's Astrophysics text and it looks to me as if you adapt stuff you are learning about general astrophysics to address habitable planet issues. It is a good strategy for learning. You let your interest in habitable planets drive your assimilation of a textbook almost on a chapter by chapter basis, it appears. Whenever you learn some general fact or equation you think of how it might apply to your special interest.

===

Also you are learning communication skills by doing this and writing it up for messageboard. Here for instance, as a communication tip, you should have put "Summer" and "Winter" in quotes!

To me it's obvious that you did not mean axis-tilt-type seasons, you meant perihelion and aphelion (near and far) seasons. For a planet with substantial eccentricity those are very important to consider! More important in some cases than tilt-seasons. Putting the terms in quotes could have made it clearer to the reader what you were saying.

 

I've noticed in the exoplanet catalogues that many of the planets that have been discovered do indeed have quite large eccentricity compared with earth. As I recall it isn't unusual to see eccentricities as large as e = 0.2!

 

You seem to have a proof here that orbital eccentricity should be included in formulas defining the habitable zone around any given star. I assume this is well-known. Now that you point it out it seems obvious. I am far from being an expert in this kind of thing so I don't know if they typically include it. I don't think this matters. Educationally your contribution can be good (both for you and us) as long as you develop the communication smarts not to excite member antagonism. Welcome, and good luck.

 

===

BTW why did you pick Bradley Carroll and Dale Ostlie's book? I don't know the book. Did you consider any others? What's especially good about C&O's text?

Wow. OK, I see. C&O is the obvious choice for an introduction to Modern Astrophysics. I checked the amazon listing

http://www.amazon.com/gp/bestsellers/books/13445/ref=pd_ts_pg_2?ie=UTF8&pg=2

It is the highest rank astrophysics textbook.

Edited March 15, 2009 by Martin

[Widdekind]

That is a fair suggestion, I would "edit" in the quotes if I could. I was indeed referring to "seasons" caused by high Eccentricty. However, the eccentricity term is, functionally, of the same form as the "Gamma Factor" in Special Relativity. The effect only becomes appreciable for very high eccentricities, e ~ 0.8+.

 

Never-the-less, Eccentricity seems worthy of consideration, for more complete pictures of Habitability issues.

 

(Carroll & Ostlie was the required text in Grad School, evidently being widely popular, w/ both Prof. and students.)

[Martin]

...(Carroll & Ostlie was the required text in Grad School, evidently being widely popular, w/ both Prof. and students.)

 

Hi Widdekind!

 

I am glad you came back and checked your thread. From some other discussion at some other board, I forget what the topic was, I connect you with UCSD.

Maybe you did some of your undergrad or graduate work there. If so you may have run into Tytler? I don't know him, just of him. He works with Keck and he seems to be interested in the habitable planet search. Maybe directly involved in it.

 

I think the habitable planet search is exciting, actually in part because it spurs the development of new instruments.

And possibly will also spur the development of new methods of inference and mathematical applications (I don't know about this but it could be.)

 

Who else is good in that line of research? Any people or projects I should keep an eye out for? Don't know much about it, just feel it has potential.

[D H]

You've done this same stuff before at other sites, Widdekind: Oversimplifying and leaping to unjustified conclusions. In this case, you are ignoring that a planet radiates. A planet with a high eccentricity will spend too much time away from its star. If the planet is Earth-like (~70% ocean), that time away from the star will make the planet freeze over, thereby drastically increasing its albedo, thereby drastically increasing the amount of incoming energy needed to make the planet hospitable for at least part of the year.

[Widdekind]

Hi Widdekind!

 

I am glad you came back and checked your thread. From some other discussion at some other board, I forget what the topic was, I connect you with UCSD.

Maybe you did some of your undergrad or graduate work there. If so you may have run into Tytler? I don't know him, just of him. He works with Keck and he seems to be interested in the habitable planet search. Maybe directly involved in it.

 

I think the habitable planet search is exciting, actually in part because it spurs the development of new instruments.

And possibly will also spur the development of new methods of inference and mathematical applications (I don't know about this but it could be.)

 

Who else is good in that line of research? Any people or projects I should keep an eye out for? Don't know much about it, just feel it has potential.

 

Small world, I TA'd for Prof. Tytler's course about basic cosmology. As for other, University of WA Prof. Woodrow Sullivan is involved in Astrobiology. U-WA also has the Virtual Planetary Laboratory (VPL).

 

 

 

QUICK RE-EXPLANATION for CLARITY:

 

Habitable Planets presumably have copious quantities of water, which is necessary for Life -- to wit, they have Oceans.

 

Oceans moderate surface temperatures. For example, the waters off the coast of Chile stay about 55 degrees F year-around*. Likewise, coastal lands have much more mild climates than places far inland (eg. Gobi Desert, Siberia)**.

*

History Channel

How the Earth was Made -- Driest Place on Earth

(TV)

**

National Geographic Channel

Naked Science -- Death of the Earth

(TV)

Thus, to make an analogy w/ food...

You can eat 2,000 calories every day...

Or, you can 20,000 calories one day, and fast for the next 9.

 

Likewise, an Eccentric planet can absorb huge amounts of energy at Perihelion, close in to the central star...

warm way up...

and then gradually cool back down, as it swings back out to Apihelion, far away from the central star.

 

Since the Oceans moderate the surface temperatures, heat re-radiation, back out to space, won't change that much, as the planet orbits. Thus, to "First Approximation", the planet's Black Body Radiation is constant, and so all that matters, for Thermal Equilibrium, is the Time Averaged Heat Influx, from the Sun, averaged over an orbital cycle.

 

So, w/ you representing our hypothetical Habitable Planet, "you" could "eat" 2,000 "calories" per "month", for 12 months...

 

Or, you could eat 3,000 "calories" for one month, 2,000 "calories" for 10 months, and then 1,000 "calories" for another month.

 

Overall, on average, that's still a "healthy diet". Sort of like Bears, hibernating for winter (0 calories eaten per day), after gorging themselves all summer (zillions of calories eaten per day). Overall, on average, the "Bears" remain lean & healthy.

 

These approximations would break down for highly Eccentric orbits (e ~ 0.8, say). But, as already noted above, of all ~300 planets known, nearly all have e < 0.2-0.3. In such a case, my simple model is valid, b/c it only makes a few percent difference, and is negligible anyway.

 

So, there's really no argument, but Eccentricity is still something to bear in mind, when considering Habitabilities of Planets.

 

 

Note: It's somewhat more complicated, b/c Eccentric orbits must have larger Semi-Major Axes, and hence longer years (Kepler's 3rd Law). So, a more accurate analogy would be:

• eat 3,000 calories for one month
  
• eat 2,000 calories for 10 months
  
• eat 1,500 calories for 2 months
  

which still Averages out to 2,000 calories per month.

Edited March 20, 2009 by Widdekind
Consecutive posts merged.

[Mokele]

You do realize that most life has a fairly narrow range of temperature tolerances, right? And that sustained temperatures above 40C will kill just about any multicellular life form within hours? Add in that most life stops working at 0C, and you have an extremely limited range of eccentricities.

[D H]

Exactly, Mokele. The primary effect of eccentricity is to narrow the habitability zone. A planet with too high an eccentricity is not habitable.

[Martin]

You do realize that most life has a fairly narrow range of temperature tolerances, right? And that sustained temperatures above 40C will kill just about any multicellular life form within hours? Add in that most life stops working at 0C, and you have an extremely limited range of eccentricities.

 

Hi Mokele,

 

my impression is that HZ has acquired a kind of consensus meaning among professional astronomers.

http://en.wikipedia.org/wiki/Habitable_zone

Wiki is often unreliable but I think it may be OK here at least in broad outline. They are saying that for a star with solar luminosity the range of semimajor axis is 0.95 to 1.37 AU

 

And for a star of different luminosity they show how to adjust the band.

 

We could do some calculations and try to see how closely the conventional HZ concept corresponds to your idea of temperature range 0 to 40 Celsius.

 

I'm not sure details like that matter though. There is a lot of latitude implied by albedo and greenhouse variation.

 

Once one finds a planet within the broad HZ range, one has to estimate the atmosphere density and think about those effects. A planet where the top of the cloud layer reflects away most most of the incoming light will be a lot colder than equilibrium black body temp at that distance from star.

 

So one should not interpret the broad HZ range naively.

 

Basically it is just one step in the process of searching for earth-like or habitable planets. The search is just getting started. They cast a wide net. HZ is a fairly arbitrary category defining the first cut.

 

Let's just say that the professional planet-search astronomers' definition of HZ is 0.95-1.37 AU, adjusted for luminosity.

 

Does anybody want to challenge that and say the astronomers should be using different bounds?

---

Merged post follows:

Consecutive posts merged

But, as already noted above, of all ~300 planets known, nearly all have e < 0.2-0.3.

 

Exactly, Mokele. The primary effect of eccentricity is to narrow the habitability zone. A planet with too high an eccentricity is not habitable.

 

DH, can you give us some idea of what you mean by "too high".

 

The point of the OP is that eccentricity needs to be taken into account, so I take it you would agree with Widdekind on that general point.

 

The critical factor seems to be the averaging over the year performed by the ocean.

 

Maybe the planet searchers are right not to worry about eccentricity very much in making the first cut. Maybe for a planet with an ocean orbiting a sunlike star the main thing that matters is simply the semimajor axis*. I don't have an immediate opinion about this. Do you?

 

* assuming say the e < 0.2.

[Mokele]

Martin, the wiki link says the center of the habitable zone is defined by being warm enough for liquid water, which is more or less where I was going, except to point out that too high of an eccentricity would bring a planet beyond this range.

 

IMHO, the important factor isn't the average, it's the minimum and maximum. If you have a planet with an average temperature of 20C, but it fluctuates between -50C and +250C, it cannot support life, as either of those conditions, especially the latter, would be lethal.

 

Oceans retain heat, but there isn't a huge variation in solar input across the whole planet like there would be for a highly eccentric orbit.

 

And remember, most of the oceans is a vast semi-empty dead zone - the vast majority of ocean biomass is near the shore and in shallow waters, where the insulating effect will be significantly less.

[Martin]

Martin, the wiki link says the center of the habitable zone is defined by being warm enough for liquid water, which is more or less where I was going, except to point out that too high of an eccentricity would bring a planet beyond this range.

...

 

Good. Now what do we mean by "too high" in some realistic case?

 

Equilibrium temp goes inversely as square root of distance. So an eccentricity of 0.2 or 20 percent translates into a 10 percent excursion in absolute temp, if unmoderated by the ocean.

 

If absolute temp is 300 K, then 10 percent is 30 K.

 

This is assuming no participation by cloudtop reflection albedo and by greenhouse. It is a really really rough envelope scribble just to get an idea.

[Ophiolite]

You do realize that most life has a fairly narrow range of temperature tolerances, right? And that sustained temperatures above 40C will kill just about any multicellular life form within hours? Add in that most life stops working at 0C, and you have an extremely limited range of eccentricities.

I shall set aside completely the possibility that alien life forms may enjoy a wider range of temperature tolerances as the consequence of a different (perhaps radically different) biochemistry.

 

There are several ameliorating influences on temperature. Widdekind makes the central point about water. Martin mentions albedo changes and is probably thinking in terms of cloud/ice cover variations, but I would add possible changes in vegetative reflectivity with the eccentric seasons. Martin also mentions greenhouse effects. I'll add, what if the alien 'vegetation' had evolved to emit significant quantities of methane in response to falling temperatures.

 

These sort of variations permit a vastly wider range of 'natural' temperatures for the HZ that will still translate into variations that fall within the range that concerns you.

 

Finally, we are talking about the habitable zone. That is not the multi-cellular HZ, but the HZ. Thermophiles would have no problems in the 40 to 80 degree range.

 

Does anybody want to challenge that and say the astronomers should be using different bounds?

No one has addressed the issue of a migrating HZ. The HZ must migrate because the luminosity of the star varies over time. If we are interested in finding planets capable of sustaining the kind of multicellular life that interests Mokele, with suitably scaly skin and cold blood, then it is not where the HZ is today, but where was it 1 billion years ago and where will it be one billion years in the future. This will seriously reduce the size of the viable zone.

 

(If I wanted to dismiss my own argument I would just mutter Gaia and be done with it.)

[Mokele]

I shall set aside completely the possibility that alien life forms may enjoy a wider range of temperature tolerances as the consequence of a different (perhaps radically different) biochemistry.

 

The problem with postulating different biochemistry is that you have a fundamental tradeoff. If you have more stable molecules, they're more resistant to heat, but they're also harder to process via metabolism.

 

I'm not saying Earth is at the optimum, or that it's not possible, just that there are very real physical constraints on what life can do which are often brushed aside by invoking the argument of "but they're aliens!"

 

There are several ameliorating influences on temperature. Widdekind makes the central point about water. Martin mentions albedo changes and is probably thinking in terms of cloud/ice cover variations, but I would add possible changes in vegetative reflectivity with the eccentric seasons. Martin also mentions greenhouse effects. I'll add, what if the alien 'vegetation' had evolved to emit significant quantities of methane in response to falling temperatures.

 

As I pointed out, I'm skeptical how much insulation water would provide life, since life tends to congregate in the shallows - structural complexity leads to more niches, more critters, more food, etc. The relative biomass of the open oceans, especially in the deeper portions, is far poorer than most terrestrial deserts.

 

Albedo and greenhouse effects could ameliorate things, but the vegetation issue is, IMHO, putting the cart before the horse - before you get vegetation, life has to originate in the first place, and those original lifeforms likely were too small in number or size to significantly alter their habitats at any appreciable level.

 

That's part of the problem - unless we speculate about panspermia, for a planet to have life it must not only have conditions to which life could adapt, but must have conditions favorable to life originating in the first place.

 

That is not the multi-cellular HZ, but the HZ. Thermophiles would have no problems in the 40 to 80 degree range.

 

And there are fish that can tolerate -20C, and small invertebrates that can completely dessicate and then revive.

 

But these are all derived traits - primitive life likely had roughly the same temperature tolerances as modern bacteria, which isn't really that far off from most other living things.

 

 

 

I think we're getting hung up on details, and missing my main point, which is that life is *not* completely unpredictable, and is influenced by some very basic and universal physical rules. The whole point of an HZ is where life is likely to occur, so if we don't take these rules into account, what's the point?

 

I'd also like the point out that speculation effectively destroys the value of the HZ - we could expand the HZ from 0.01 to 1000 AUs if we got creative enough about potential life forms, how to define life, etc. Hell, if we speculate enough, we could have life evolving in interstellar gas clouds, making the HZ basically anywhere that isn't inside of a star or black hole.

 

I'm not saying life on a planet with an eccentric orbit is impossible; I'm saying that there are real limitations to life that are being overlooked in the current analysis, such as the lethal effect of certain temperatures regardless of the year-long average temperature. These can be ameliorated, as others have shown, but how much speculation about ameliorating factors is allowed?

 

Is life on a highly eccentric planet possible? Sure. Is it possible outside the range of temperatures seen in the usual HZ? Sure. But based on our planet, as well as the underlying rules of life, planets in the HZ are more likely to have life than those outside, and those with less eccentric orbits are more likely to have life than those with high eccentricity.

[CharonY]

I guess it would be beneficial to first take out the "habitable" element out, as the discussion is about two things, really. So basically one point would be just to define what influences eccentricity will have (as it basically has been done) and maybe add more relevant variables.

 

If we really want to discuss habitable in this context we have to restrict it to what we know about organisms (as Mokele pointed out). Otherwise the denominator "habitable" has to be stretched to a point where it ceases to have a meaning (essentially being a tautology, inhabitable for those with the modifications required to inhabit it).

[Martin]

... Otherwise the denominator "habitable" has to be stretched to a point where it ceases to have a meaning (essentially being a tautology, inhabitable for those with the modifications required to inhabit it).

 

There was a lot of discussion in the past.

Now HZ has a conventional technical meaning.

Money and telescope time is being allocated to the research goal of finding HZ planets.

 

Another term that seems to have taken on a practical operational meaning is "earth-like planet".

 

Careers and resources are being allocated to the "earthlike planet search"

 

The technical meaning of these terms is kind of pragmatic. You have to set your goals and define your criteria. And that has been done, because then science policy has to be framed and work has to go forward.

 

HZ means 0.95 to 1.37 AU adjusted for the star's luminosity.

Earthlike as I understand means something like a rocky planet with water that is no more than 6 or 7 earth masses and orbits in the HZ.

I'm not an expert in this, Widdekind may know the agreed-on definitions.

 

I think we've had enough of these discussions about "what is Life?" and quibbling about terms like "what does habitable mean?"

Now the game is to design and launch instruments that can identify planets in the interesting band and find out as much as we can about them.