Star mass/luminosity for a given HZ

[Ian J.]

Hi,

I've been doing a bit of reading of various online sources to try and understand how to calculate the star type for a Habitable Zone for a planetary system I've already got an orbit distance for, but my poor maths can't cope (at the moment) with the equations.

Can someone point me in the direction of a way of calculating the mass, diameter, temperature and luminosity of a star to give a Habitable Zone centre of approximately 93 AUs?

TIA

Ian

 

[mfb]

I would look up tables for known stars.

93 AU is problematic. This requires a very luminous star (~10000*sun), and those are very short-living. In addition, they emit a lot of UV radiation, which is bad for the stability of chemical bonds.

 

[Ian J.]

I've been mulling over my question and I think I was a bit premature with it, and your answer confirms my suspicion.

So, I will take on board what you've said and check out known stars before I try and rephrase the question. Certainly the 93 AU now seems silly. I need to reduce the mass of the star orbited and the orbit distance of my invented world to something workable within reasonable known observations.

 

[Ian J.]

I think I might understand what I'm trying to find out now, so my question becomes:

For a star of any given mass, what are the factors that influence the distance and breadth of the Habitable Zone, where that HZ is suitable for humans to live comfortably?

I'm thinking of a lower numbered A class star (say A1 or A2) for my invented system, but that might not be the right one. What things in an A class star might make habitability (for humans) difficult?

 

[mfb]

Well, there is a very simple requirement: the incoming light intensity has to be similar to earth. This is proportional to the stellar luminosity divided by the squared radius:
To get the same intensity at 2 AU, the star needs 4 times the solar luminosity (as intensity drops with 1/r^2).
The precise borders of the zone depend on your favorite method to evaluate the possibility for humans to live there.

Stars which are too luminous (this includes class A) are short-living - but if you want to move humans there, this is not an issue. Millions of years are very short for evolution, but ages for humans. Those stars are often very hot (-> emit a lot of UV radiation), so UV protection might be necessary.

Stars which are very dim have their habitable zone close to the star, where planets are forced into bound rotation - they do not have days any more. This leads to large temperature differences and winds between the "day hemisphere" and the "night hemisphere".

 

[Ian J.]

That's a good point regarding the luminosity needing to be similar to that of Earth.

I have found an online calculator at http://depts.washington.edu/naivpl/sites/default/files/HZ_Calc.html that seems to be reasonably up to date with current thinking (it references "Habitable Zones Around Main-Sequence Stars: New Estimates" by Kopparapu et al.(2013), Astrophysical Journal, 765, 131 which seems to be this year) and I'm currently using it to cook up some ideas.

I think the relative short time that an A series star holds it's luminosity could be a problem for me using one of those spectral types, as I need an evolutionary time scale for stability. That suggests an F instead as the likely largest type based on my 'studies' this afternoon. However I'm not sure an F would be massive enough or luminous enough for what I have in mind.

Unfortunately, in previous development and writing I have left myself with an awkward orbital period of 39.36 years for a sub-system orbiting the star, where the sub-system has several moons of similar size to Earth orbiting what is probably either a brown dwarf of some kind or an 'at limit' massive gas giant, in a scenario which suggests perhaps a smaller system 'caught' by another. I'm trying to resolve the problem of the star size into possible, if improbable, known physics without having to completely rewrite the existing scenario. The A series star in the lower number order seems to allow me to get away with retaining that orbit period, but I'm now finding it's too short lived.

Back to the drawing board...

 

[Bandersnatch]

That's a good paper and associated calculator that you've linked to. It takes into account many more factors than just the intensity of radiation, including greenhouse effects. (I don't think it's meant to be used with stars hotter than the F class)
The calc is a bit clunky to use, though. You might want to use this one instead:
http://astro.unl.edu/naap/habitablezones/animations/stellarHabitableZone.html
While it doesn't appear to be as accurate as the other one regarding the habitability range, it is much more interactive(and fun).

Also, you might find this poor-man's numerical integration calculator to be of some use when trying to concoct a stable system:
http://www.orbitsimulator.com/gravity/articles/what.html

If you tell us what you're looking for in more detail, we might be able to help you a bit more.

 

[Ian J.]

In a reply to a previous post of mine (here: https://www.physicsforums.com/showthread.php?t=703405) Gravitation3D was suggested to help with n-body stuff, but I will look into the Gravity Simulator as well. I'm using another piece of software for my basic system construction (AstroSynthesis) but it's not got proper gravity or degeneracy pressure simulations so allows some really silly values to be used for sizes of stars, planets, etc, which is why my knowledge of astrophysics needs to be better to prevent me making silly mistakes.

I like that habitable zone simulator, if only for the face on the planet after the star has 'died'! :D

It's late here in the UK, so I haven't time to go into more detail other than at the end of my last post, but tomorrow I'll try and reply with a better explanation of what I'm trying to do.

 

[Ian J.]

I think the best way I can describe what I'm looking for at the moment is to provide a bit of background:

I wrote a couple of novels in 2000 and 2008 (the beginning of a series) that established a star system where the orbit of a particular world is 39.36 Earth years. I have since found out that various elements of the system as I envisaged it don't hold up to reasonable scientific scrutiny. I am currently approaching doing new editions of the novels to spruce up the text and fix some things I got wrong, and I want to take the opportunity to fix the science errors as best I can while I'm at it.

There are two things that I don't really want to change within the created system:

Thing 1: There is a large world around which several moons of Earth mass, size and type orbit. They need to stay in place. The crucial factor in two of the moons is that they are in a 180 degree co-orbit, and their orbit period is 409.08 Earth days. I have constructed some elements in the calendars of the societies on those moons that depend on that value staying constant, so changing it would be a pain.

Thing 2: Following on from Thing 1 above, the orbit of the large world is 39.36 Earth years around the star. This also plays a part in some of the societal development, so again I don't want to change that if I can possibly avoid doing so.

As far as I can see, there are two things that need fixing:

Fix 1: The large world for the moons needs to be better defined. I originally had it as a gas giant, but I don't think it would be massive enough, so currently I am thinking it probably needs to be a brown dwarf of some kind. I'm not sure what it's mass could realistically be, but currently I have it at about 60 Jupiter Masses.

Fix 2: I need a decently massive, stable, longish lived star of a type with a luminosity that creates a habitable zone the centre of which is at the point where an orbit of 39.36 Earth years sits. The distance from the star to the middle of the HZ is less important than the need that anything that is orbiting in the centre of the HZ has an orbit period of 39.36 Earth years. This isn't a binary system, so the large mass of the star is relatively important in order to offset the size of the large world / brown dwarf so that the barycenter isn't too far away from the star. I'd prefer it to be as close to the star as possible, if not inside it.

I'm aware I may be setting myself up for other problems, going by some stuff I've been reading online. Things like tidal locking need to be avoided, and I also need to be careful of the Hill spheres for both the star and the large world (and for other planets in the system too).

Is this enough to go on?

Reading the Wikipedia article on 'Circumstellar habitable zone', mention is made of 'Upsilon Andromedae d' being a candidate for earth-like moons, so I think what I'm envisioning isn't outside the realms of possibility, even though it might be highly improbable.

That fits in with a 'mantra' I'm developing, which is that I want the science for my 'space opera' stories to be possible as much as it can be, even if it's highly improbable at times. I might have to push limits occasionally, and even do the dreaded FTL travel eventually, but I'm trying to be as close to known and possible science as I can rather than just ignore science altogether and create junk backgrounds for my stories that won't hold up to scrutiny at all.

 

[mfb]

Thing 1: There is a large world around which several moons of Earth mass, size and type orbit. They need to stay in place. The crucial factor in two of the moons is that they are in a 180 degree co-orbit, and their orbit period is 409.08 Earth days.

That is an unstable orbit - it won't exist for an extended period of time.

I originally had it as a gas giant, but I don't think it would be massive enough

I don't see an issue here.

A brown dwarf could be an interesting method to shift the habitable zone outwards. Let the gas giant provide infrared radiation to keep the planets warm, and get some light from the distant star.

so that the barycenter isn't too far away from the star.

Why? What's wrong with some motion of the central star? This will not even be notable until your inhabitants make precision measurements of stellar positions.

I'd prefer it to be as close to the star as possible, if not inside it.

Even Jupiter violates this, and it is closer to the sun.

 

[Bandersnatch]

Re: thing2, i.e., the orbital period being 39.36 years. For a star of 1 solar mass, the orbit lies at 11.5 AU, and increases as $\sqrt[3]{M}$. The habitable zone "catches up" with the orbit(at a bit over 18AU) for a stellar mass around 4 solar(using the interactive calculator - the one by Kopparapu et al. is not made to be used with such bright stars).
This star would have a very short lifespan, in the order of hundreds of millions of years. This means no native life forms. I'm not sure if that's even enough for the planets to get past the molten crust phase.

You might want to go with a brown dwarf as the other large body, to supply the bulk of the heat instead - as suggested by mfb.
However. Even if you were to take a large brown dwarf similar to Teide 1(http://en.wikipedia.org/wiki/Teide_1), with around 60 Jupiter masses and ~2700K temperature, you end up with the habitable zone extending between 0.03 and 0.07 AU. For the moons, this corresponds to an orbit of 25 days, rather than the required ~409. You could extend it a bit as part of the energy comes from the primary star, but you won't be able to marry both periods(39ys & 409 days) with this setup.


Bottom line is: you need to sacrifice something if you want scientific plausibility. Either shorten the period around the star, shorten the period around the gas giant/brown dwarf, or turn the latter into a full-fledged dwarf star(I think ~0.8 solar masses would do, with greenhouse effects and part of the energy supplied by the faraway larger star).

 

[Ian J.]

mfb & Bandersnatch:

Thanks for the replies, I needed to know the info, even though I obviously have some re-thinking to do!

Certainly both the orbit lengths and the co-orbit are a problem. The orbits and the nature of the larger planet / brown dwarf / dwarf star are things I could rework, though it will cause me much calculation pain and a not insignificant amount of shift in some story elements

The co-orbit is the bigger problem for me, due to some story elements being dependent on it, so that's going to be a very difficult one to solve

 

[mfb]

I played around with Excel (-> attachment), and I cannot find a useful combination - if the length of a "month" and year are right, the received luminosity is always too low, unless the lifetime of the main star is below 1 billion years or the brown dwarf gets exchanged by a sun-like star - but then this would be the main sun for the planet.

Yellow fields are inputs, the columns to the right of the stellar parameters can be used for templates for other stars. The password is empty, feel free to modify it if you like.

 

[Ian J.]

Thanks. I will keep the Excel workbook in mind while I think through what I have to change in order to get a working system. I can see myself doing some major changes, but the idea of a low power dwarf sun in place of the planet might end up being the way I go. Whatever I decide, be it that or changes to the orbit periods, the changes are going to require some re-wiring of the way the societies on the moons function, so it's no small reworking either way. The biggest issue remains the co-orbit problem. I'm going to have to think long and hard about how to alter the story to get round that.

 

[Ian J.]

OK, I have worked out possible change to my invented system to 'avoid' the co-orbit problem, but it's not a change I really want to do.

So, before I commit to it, I want to fully understand why a 180° co-orbit is problem. Can anyone fill me in on the reasons for not doing it, using, as much as possible, laymen's terms? The only online works I can find on co-orbits use copious amounts of formulas that leave my brain feeling like mashed jelly.

Also, if a simple 180° co-orbit isn't stable is there any, and I mean any, way to set up a system to allow a 180° co-orbit to be stable? My thinking in asking such a question is that I wonder if it might be possible to introduce another body / other bodies into the system (but not in the same orbit) in such a way that the two 180° moons orbiting the central world remain stable.

TIA

Ian

 

[mfb]

Also, if a simple 180° co-orbit isn't stable is there any, and I mean any, way to set up a system to allow a 180° co-orbit to be stable?

Not without active control mechanisms - technology from some intelligent species.
Any deviation from a (mathematical) perfect alignment leads to an amplification of that deviation, the motion of the particles becomes chaotic soon.

• Separations of 60° are stable, if one object is much smaller than the other (something like a factor of 25 difference in masses).
• Horseshoe orbits are stable and reach the 180°-separation from time to time, but not always and they still need one big and one small moon.
• Epimetheus and Janus, two moons of Saturn, have a very special orbital configuration - but no stable 180°-separation...

Why do you need those 180°? Do you want the moons to be invisible from the other moon?

 

[Ian J.]

Not without active control mechanisms - technology from some intelligent species. Any deviation from a (mathematical) perfect alignment leads to an amplification of that deviation, the motion of the particles becomes chaotic soon.

Hmm. I've just had an idea. It fits in with something only vaguely hinted at that I had already written in the first novel, but I could use it to hold the moons in co-orbit. It won't be apparent in the first novels in the series how it works, just that it does. Explanations would have to come much, much later on.

Why do you need those 180°? Do you want the moons to be invisible from the other moon?

Kind of. It's not so much that they are entirely obscured, just that one moon is more difficult to reach than another. The society on moon B is the most advanced, and having just reached space exploration capability, albeit with a different technological arrangement than us on Earth, they have to make a choice which of two moons A and C they are going to land on. Moon A would be the obvious candidate if they could study it in any detail, but as it's partially obscured they go to the easier to observe and reach moon C instead. This leads to a pattern of events that suits the story I want to tell. If they could study moon A in any detail, they'd go there first without a doubt, and that would be a sufficiently different story that it wouldn't be the one I want to write.

 

[Bandersnatch]

So, does your story have the spacefaring civilisation move straight to manned spaceflight without first sending probes around? They sent a manned mission without first finding out what to expect on arrival?
Because if they did send probes, then there's nothing difficult in reaching a planet 180 deg away on the same orbit. Probably easier than any other planet.

 

[mfb]

A co-moving moon is very easy to visit with probes due to its similar orbit. Moons on a completely different (or even retrograde) orbit would be more difficult to visit.

 

[Ian J.]

Hmm. I haven't got answers to those points, as I hadn't thought about it in that much detail. This is the kind of thing that my 'common sense' thinking fails on.

I did think society B would have sent some kind of basic small craft out, but they wouldn't have been much more advanced than a Sputnik 1 other than having some camera equipment into take pictures. I think they just see it as easier to send a manned mission sooner and moon C is a more visible candidate to them than moon A. Their probes to A, for whatever reason (I don't know yet) won't have given them enough information to act on.

If all else fails and this arrangement just isn't plausible enough, then I can go to my alternate though less liked plan of changing moon A into a planet and putting it into a habitable zone closer to the star. In that way it's so far away from moon B that it would be like them trying to get to Mars instead of the Moon the way we currently are on Earth. But I have other story reasons for not wanting to do that.

 

[mfb]

Well, I think a manned (or whatever your inhabitants are called) mission without some basic exploration of the moon system is very unrealistic. If moon A and C look very similar, different orbits are a good argument to go to one of them first. If one of the moons looks way more attractive, it gets harder to find arguments.
If A has life, maybe the inhabitants of planet B don't want to mess around with the ecosphere of A?

 

[Ian J.]

Agreed re basic exploration, though I'm not sure how much. I did a bit of looking up on the various missions by US and Russian space programs to the Moon before a manned attempt was made, and also the attempts to get a probe to Mars. Seems in those early days there were quite a few failures, but surveys were attempted and eventually done, so some level of research exploration would be done. I need to think about how that impacts the co-orbit vs the non-co-orbit variations and see how I feel about the best approach story-wise while still retaining the essence of the story I want to tell.

Both A and C are inhabited, so there would be no issue there, it's just that A is more advanced than C and given equal exploration of both, I think the Bs would go for A first, not C. But unfortunately I can't backtrack now and change some essential elements of the story so B must go to C first.

 

[Bandersnatch]

If the inhabitants of C are advanced enough to have radio/tv etc., then emissions could be easily detected at B, prioritising the effort to get there rather than anywhere else, especially manned missions. You could have the public living for a few good decades with the knowledge that there are sentient beings on C, dominating the imagination of everybody on the planet, driving the policies and government funding, etc.

Now you'd just need something to interfere with similar emissions from A. If you're going with a star to replace the original gas giant, then maybe it's especially active in radio frequencies or somesuch thing that would prevent reception of radio signals from the planet more or less behind it?

Something else than radio waves could serve the purpose(like the canals of Mars, only actually real). The point would be to make the people demand going there as soon as possible to "meet the aliens".

 

[mfb]

It is extremely unlikely that 3 independent civilizations independently reach the technology for radio transmission within a few decades.

If both planets are inhabited, but without radio technology and similar technologies, it might be hard to evaluate their technological levels from space. They are (probably) completely different species with a mainly independent evolution. Sure, a common origin of life in the system is reasonable, but you cannot transfer more than some lonely cells via asteroid fragments (I have some doubts that this would be sufficient to transfer complex multicellular organisms... but it is science fiction, so it is fine).
You could easily choose the "wrong" planet to go first.

 

[Ian J.]

Neither A or C have technology for radio transmission, so that's not an issue. A has a kind of Roman/Renaissance/Pre-Victorian feel, while C is more of a Dark Ages/Pre-Medieval feel. C as I mentioned is at early spaceflight stage. There is a link between them that answers the independent evolution question, but that's not something that becomes clearly apparent until later in the first sequence of books.

 

[Ian J.]

I am not sure, but I think I am likely to settle on the main star plus dwarf star idea, with Moon A converted into Planet A and orbiting closer to the main star in a suitable HZ there, and have Moons B and C around the dwarf star further out with suitable orbits for the mass and luminosity of dwarf.

I think I can retain the calendar of A without too much difficulty, but just lose the 'months' as there wouldn't be a moon of any significant size (tides would not be strong, but might exist in a narrower range based on the main star gravity?).

The orbits and calendars for B and C would need to be adapted. At the moment B shares its orbit period with A, but I think that having both A as a planet and B as a moon with the exact same orbit period would be pushing plausibility further than I'd like. C fortunately hadn't had any calendar planning in detail so that's easy to change, B is a lot more work unfortunately.

So, what are likely to be the pitfalls of having Planet A in a closer HZ at 409.08 Earth days orbit period, and Moons B and C around a dwarf star with not dissimilar orbit periods (say 350/500 or so Earth days)? What kind of star is the Main star likely to be? What kind of star is the dwarf star likely to be? I think Dwarf would need to still be in main sequence and not a White Dwarf or whatever?

TIA

Ian

 

[Ian J.]

Neither A or C have technology for radio transmission, so that's not an issue. A has a kind of Roman/Renaissance/Pre-Victorian feel, while C is more of a Dark Ages/Pre-Medieval feel. B as I mentioned is at early spaceflight stage. There is a link between them that answers the independent evolution question, but that's not something that becomes clearly apparent until later in the first sequence of books.

I've just seen an error in this message - 'C as I mentioned...' should read 'B as I mentioned...'

 

[mfb]

Well, A and B can have the same orbital period. If they do not have that, as reasonable separation (factor 1.5 or more for the periods) should be fine as well.

What kind of star is the Main star likely to be? What kind of star is the dwarf star likely to be? I think Dwarf would need to still be in main sequence and not a White Dwarf or whatever?

Main sequence star.

I think I can retain the calendar of A without too much difficulty, but just lose the 'months' as there wouldn't be a moon of any significant size (tides would not be strong, but might exist in a narrower range based on the main star gravity?).

A can get a moon :).

 

[Ian J.]

The orbit period could be the same for A and B, but I think of it as so highly improbable that it would be less credible and acceptable to readers. I am thinking the orbits could/should be similar, but not the same.

What type of Main Sequence star? I'm wondering about spectral type, mass and luminosity. How big can it get before it's size would have significant consequences on the HZ and orbit period? Also, how small can the dwarf star get before it wouldn't have sufficient mass to be a star at all?

Re moons around A, I'm trying to avoid having A be a total Earth clone - I want it to feel different in enough ways but still be plausibly a place where human-type life could comfortably live. So I'm thinking either no moons, or some very small, less influential ones.

Once again, TIA, all this help is much appreciated, even though it gives me headaches of problems to solve relating to my story :)

 

[Bandersnatch]

Here you go:
https://docs.google.com/spreadsheet/ccc?key=0AqhJly6aYB0ZdFVKZjV1YW9fR3k2MGpTV2MzT1JLY0E&usp=sharing

Input data in the green cells. This includes the "mass" cells, although it is unlikely you'll need anything beyond the range already listed, as the long orbital periods you prefer require a specific range of stellar masses to work.

The blue cells show the range of habitability for a given orbital period.

The coloured cells of the stellar classification column roughly represent the actual colour of the star, although you should keep in mind that in reality these would all be just various shades of white. There's a relevant discussion regarding the subject of stellar colour here: (https://www.physicsforums.com/showthread.php?t=705655).Luminosity follows the mass-luminosity relation of $L\propto M^{2.5}$ for $M<1$ and $L\propto M^{3.88}$ for $M>1$

Stellar radius: $R\propto M^{0.57}$ for $M<1$ and $R\propto M^{0.8}$ for $M>1$
(http://www2.astro.psu.edu/users/rbc/a534/lec18.pdf)

Temperature uses $T\propto \frac{L^\frac{1}{4}}{R^\frac{1}{2}}$ (http://zebu.uoregon.edu/~soper/Stars/trl.html )

Stellar classification is assigned according to: http://en.wikipedia.org/wiki/Stellar_classification#Harvard_spectral_classification
Remember that you're interested in main sequence stars, so all will include an additional Roman "V". You might want to include arabic numerals between 0 and 9 to mark in which tenth of the temperature range for a given letter the star lies in(e.g., the hottest A stars between 10000K and 9750K would be fully denoted A0V).The planet's orbital radius is calculated from Kepler's 3rd Law.

Insolation follows inverse square law, and the range of habitability is taken from the Kopparapu et al. calculator.
This is obviously a simplified approach. For example, luminosity depends not only on mass, but also on age and metallicity, and stars are not ideal black bodies. But perhaps that's enough of accuracy for your needs.