Tidal implications of eccentric planetary orbit

onomatomanic

I'm world-building for a SF-story and need some help figuring out what the ocean tides would look like.

My planet orbits a close binary star system; it does not have any satellites itself. The mass of the binary is 5/3 Mo [solar masses]. The orbit is closer than that of Earth and is highly eccentric (a ~ 1/2 AU, e ~ 4/5), such that the planet orbits at about 0.1 AU at perihelion and at about 0.9 AU at aphelion. The value of e was chosen for plot purposes, while the value of a is simply a result of the requirement that the planetary insolation, averaged over the year, must be similar to that of Earth. The planet rotates once every 60 (Earth-)hours.

Originally, I simply assumed that lack of moons meant that the only aspect of tidal effects I needed to worry about was long-term, namely tidal resonance. According to the 'pedia article, the timescale for tidal locking scales as a^6, meaning that my planet would be locked to its binary 64 times more rapidly than Earth to Sun*. Given that Earth is nowhere near locked, and that I'm free to posit that other parameters which come into play, such as the planet's angular momentum upon formation, would act to delay locking, I should be fine in that regard: It's not implausible for my planet to have remained unlocked for long enough for my protagonists, a human-like species native to the planet, do have evolved.

But that initial assumption was nonsense, of course - short-term tidal effects such as ocean tides do have to be carefully considered, moons or no moons. On Earth, solar tides are only slightly (less than an order of magnitude) weaker than lunar tides. The only reason we don't ordinarily notice the former is that they manifest as modulations of the latter, rather than as effects in their own right. So, even around aphelion, my planet would experience ocean tides of a magnitude comparable to those on Earth.

Where things get... interesting, let's say, is around perihelion, however. Tidal forces scale as M/d^3, I believe, where M is the mass of the force-exerting body and d the distance between it and the force-subjected body. Scaling my case to the Earth-Moon one, that gives

F_tidal ~ (M_binary/M_Moon) / (d_aphelion/d_Moon)^3 * F_tidal_Earth
F_tidal ~ ((5/3 * 2*10^30 kg)/(7*10^22 kg)) / ((0.1 * 1.5*10^11 m)/(4*10^8 m))^3 * F_tidal_Earth
F_tidal ~ 10^3 * F_tidal_Earth

I'm not certain, but I think an increase in tidal force by a factor of one thousand means an increase in the height of the tidal bulge by a factor of one thousand. On Earth, the characteristic amplitude of the tides is on the order of a metre ('pedia), so on my planet, that would mean a characteristic amplitude of a kilometre. Not what one would call negligible.

Okay, that's as far as I've progressed. If someone could check my work and point out any flaws, that would be appreciated. Mainly, though, I need help getting a handle on the final step - translating the abstract idea of "kilometre-high tides" into a mental image of what those would actually do to a planet like Earth. For starters, I'm looking for answers to questions like the following:

• On Earth, tidal amplitudes vary considerably from place to place, by as much as an order of magnitude. Would the same apply in my case, i.e. would there be some coastlines where the dreaded perihelion-tide would "only" reach a few hundred metres above normal, and others where it would be not just one but several kilometres? Or would the much greater base amplitude swamp out those contributions which give rise to the differences observed on Earth, so that there'd be little noticeable difference in my case?
  
  
• On Earth, it doesn't seem to be particularly relevant how a coastline is oriented with respect to the planet's rotation. The tides experienced on East-facing coasts aren't noticeably dissimilar to those experienced by West-facing coasts, nor to those experienced by North- and South-facing coasts. Rather, what matters is the size and shape of the body of water bounded by the coast in question. Would this be the same in my case, or would the question of whether a given coastline represents a "leading edge" or a "trailing edge", in terms of planetary rotation, result in significantly different types of tides?
  
  
• Would a kilometre-high bulge basically behave like a metre-high bulge, i.e. would it simply flood the land which is less than its height above normal sea-level, and then recede? Or would it be more like a wave on a beach, which has the ability to "climb" a shallow slope quite a long way above its own height, due to inertia? In the former case, highlands should be habitable for land-dwellers, while in the latter, the bulge could conceivably simply keep going across an entire continent and re-join the ocean on the other side - so land-dwellers, if they'd exist at all, would have to have a way to deal with flooding no matter where they lived.
  
  
• How violent would the impact of the bulge on a coastline be? Should one be thinking Earth-like flood, scaled up, or rather mega-tsunami? On the one hand, a rise in sea-level by a kilometre over 15 hours (a quarter of one of my days) translates into a rate of a few centimetres per second, which sounds quite benign. On the other hand, the speed of the bulge with respect to the surface is the same as the rotation speed of the surface itself, which is on the order of a thousand kilometres per hour, which doesn't sound benign at all. I suspect neither of those two figures is particularly useful, though, except as maybe some manner of upper and lower bound.
  
  
• Would it make a significant difference whether the geography of my planet is water-dominated (i.e. has a number of isolated landmasses emerging from one contiguous ocean, as is the case on Earth) or land-dominated (i.e. has one contiguous landmass containing a number of isolated oceans, rather like really big lakes)? I'm thinking that the latter should make the tides somewhat less extreme, as any one body of water would never experience the planet's full tidal differential.

Any pointers would be appreciated. Needless to say, seismic effects should probably also be taken into consideration in all this, but I ultimately feel more comfortable ignoring those for the sake of plot development, if it comes to that. In that sense, the ocean tides are the primary concern, for the time being.

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* That's ignoring tidal resonances between Earth and Moon, which is silly in general but should hold for this line of argument specifically, I think.

mfb

Before you work out details of your tides: How can life evolve at the surface with e=4/5? The intensity of the solar radiation varys by a factor of 81 between perihelion and aphelion. The peak intensity would be close to 100kW/m^2, or equivalent to a good lense in bright sunlight, used to burn stuff.

I can imagine life deep in oceans (if they are not boiled+blown away in "summer"), where the seasons average out somehow, probably living from hydrothermal vents or similar stuff. But on the surface? Really?

The height of your tides will depend on the depth of the oceans and their global distribution. Apart from the obvious limit (you cannot displace more water than you have), tidal forces work best with a lot of water available (=> lower water velocity required)

onomatomanic

The total luminosity of the binary is about 0.1 Lo. So, near aphelion (0.9 AU), the intensity would be in the 100 W/m^2 range, not the Earth-like full kilo-Watt which I'm guessing you assumed. The peak-value at perihelion (0.1 AU) is "only" about 8 kW/m^2, correspondingly. Furthermore, per Kepler's second law, the planet moves exceedingly fast near perihelion, and thus the final stages of approach and first stages of recession happen rather quickly. Averaged over the one day during which perihelion occurs, the value comes down to a "mere" 4 kW/m^2 (says my numerical integration).

Admittedly, I did not really consider the question of whether those power levels might already be high enough to cause localized heating to dangerous (combustion, boiling, denaturation, ...) temperatures, as would be the case in the lense comparison you draw. What I did concentrate on was the amount of energy such a short super-summer would dump into the ecosphere, under the assumption that the energy could, in fact, be dissipated at that rate. The result I got was an increase in temperature from about 0 degrees to about 30 degrees during a three-day period centered on perihelion, at temperate latitudes. This boost is what then keeps the comparatively long (Kepler-2 again, slow motion near aphelion) winter from becoming too bitter. The reason that insolation changes which are quite a bit greater than those typically experienced on Earth (from a different source, axial tilt) don't cause even greater temperature changes than those I quoted is that the year is only about a quarter as long as Earth's, which makes buffering more effective.

If you're not convinced that this could work, have a look at this article. As to the question of how much of a problem even brief exposure to the peak insolation would constitute, I'd assumed that it'd merely mean that my flora and fauna would have to be a little more hardy and less flammable than Earth's. Do you think that sounds plausible, given the values above, or do I need to look at this again more closely?

Ah, yes, that's a good point. If an "ocean" is only a hundred metres deep, say, it'll clearly have trouble producing a kilometre-high bulge. Instead, I'm now imagining a configuration in which such a body of water would end up carving out two deeper basins, connected by a shallower "channel", and the tides would consist in the entire contents flowing back and forth between the basins, depending on their current orientation with respect to the force axis. But that's probably too fanciful... :)

D H

No it isn't. For stars of about a solar mass, luminosity is roughly proportional to mass to the fourth power. This means your total luminosity is at least 0.9645 L_sun, and that is only for the case of the two stars being of equal mass. If one of the two stars is 16% or more massive than the other the total luminosity will be greater than that of the Sun.

This is just the start of your problems. It is extremely dubious that a planet orbiting a single star with e=4/5 would be habitable. I'd throw your story away then and there. Changing the star to a close binary pair makes matters worse, much, much worse. This planet is not stable. It will either collide with one of the stars or be ejected from the system.

onomatomanic

The primary is an orange dwarf, spectral type K6, with a mass of 3/5 Mo and a luminosity of 0.1 Lo. The secodary is a white dwarf, spectral type DC6, with a mass of 16/15 Mo and a luminosity on the order of 10^-3 Lo. The previously quoted totals follow transparently.

The idea is that the planet survived the end-stages of the secondary's main-sequence evolution charred but intact, and that life processes began sometime after that. I don't know how realistic this is, but it sounds plausible enough for my purposes.

Regarding habitability, "dubious" will do just fine. A planetary environment somewhat more hostile than that of Earth is part of the high-ish concept premise I have in mind. Orbital instability is one of the problems which fall into my "ignorable" category. Nevertheless, it should be instructive to investigate this, as I'd simply taken the stability of a Keplerian orbit on faith, binary or no. Could you explain the background (if it's relatively simple) or point me to further reading (if it's more complicated)? Thanks.

onomatomanic

Okay, I think I have a tentative handle on this thing now. The basic idea is to, firstly, figure out the energy content of the tidal bulge, and then to compare that to the energy contents of, secondly, a surface wave, and, thirdly, a tsunami. Needless to say, the two latter phenomena are rather different from the former, and somewhat different from each other as well. However, other than "terrestrial tides scaled up by a thousand", which doesn't really tell me anything at all, they are the only benchmarks within human experience that I can think of.

Tide: I'm going to estimate the energy content of the tidal bulge by extracting a scaling relation from the tidal locking timescale mentioned in the OP and plugging in observational data concerning the tidal braking of Earth's rotation.

T_lock ~ p * w * d^6 / M^2
T_lock ~ p * w / F_tidal^2

where p collects the various physical constants and planetary parameters and w is the planet's rotation rate, two pi per planetary "day" with respect to the tide-inducing body in question. For order-of-magnitude purposes, the timescale of a monotonous process like this can be written as the ratio of the changing quantity, i.e. the planet's rotational energy E_rot, to its rate of change, i.e. the rate of work done by the tides, P_tidal. Using all of the above and solving for P_tidal, that yields

E_rot|P/P_tidal|P / T_lock|E ~ p_P/p_E * w_P/w_E / (F_tidal|P/F_tidal|E)^2
P_tidal|P ~ T_rot|P/T_rot|E * (F_tidal|P/F_tidal|E)^2 * E_rot|P / T_lock|E

where the subscripts to my Planet and Earth, respectively, the ratio p_P/p_E vanishes because the two planets are physically similar, and T_rot is the tidal day-length. T_rot|P turns out to be about 120 hours for the solar day of perihelion, twice its normal value of 60 hours (see OP), because orbital motion counteracts rotational motion (see diagram) and the orbital speed increases to a point here at which this makes a very noticable difference. The ratio of the tidal forces is known from the OP and T_lock|E can be extrapolated from the values given in the article linked above, leaving E_rot|P to be calculated in the standard way with the simplifying assumption of a homogeneous mass distribution:

E_rot|P ~ 1/2 I w^2 ~ 1/2 (2/5 M_P R_P^2) ((2 pi)/(60 hours))^2
E_rot|P ~ 3*10^28 J

P_tidal|P ~ (120 hours)/(24 hours) * (10^3)^2 * E_rot|P / ((24 hours)/(2 hours per 600 million years))
P_tidal|P ~ 10^18 W

A round exawatt, apparently. For reference and whatever it's worth, another wikipedia article gives the mechanical energy capacity of Earth's oceans (currents + tides + surface waves) as on the order of a measly terawatt, whereas Earth's total insolation is on the order of a tenth of an exawatt. So, this is a lot, but not quite an off-the-charts lot.

To be continued; comments would be appreciated.

onomatomanic

Surface wave: The wave power formula expresses the energy flux of a wave-crest in terms of its dimensions, height H and length L, and period T_wave. Solving for H, with L set to the circumference of the planet, that gives me

P_wave ~ 1/(64 pi) rho g^2 H^2 L T_wave
H^2 ~ 30 P_wave / (rho g_P^2 R_P T_wave)

where rho is the density of water and g the planet's gravity.

There are two obvious ways to make this comparison, it seems to me. One is to equate the energy content of the tidal bulge to that of a single surface wave travelling at the same speed, in which case P_wave and T_wave are simply the tidal power and day-length obtained above:

H^2 ~ 30 * (10^18 W) / (10^3 kg/m^3 * (10 m/s^2)^2 * 6*10^6 m * 120 hours)
H ~ 10 m

That is, the impact of the 1000-metre perihelion flood is comparable to that of a 10-metre surface wave travelling at the planet's rotational speed, which is "only" about 100 m/s due to the longer solar day - similar to today's high-speed trains. Now that, at last, is something I can work with. On the one hand, that is definitely not enough to allow the water to "climb" mountains and cross entire continents, as I originally feared. On the other hand, it is definitely enough to do some major re-arranging of surface features, in the vein of flatting a small forest and maybe even smashing a small cliff. However, since this flood happens at least once a year, there won't be any forests and cliffs to be flattened and smashed in the tidal region, so what we're really looking at is some major erosion. Every once in a while, that should lead to more spectacular things like breaching a chain of hills at the landward edge of the tidal region and flooding the valley beyond for the first time. Which is just like a broken dam writ large, and therefore conceptually straightforward.

The other way is to spread out the energy content across evenly spaced wavefronts with a T_wave on the order of what is typical for Earth's surface waves, which, from the examples given in the link above, means about ten seconds. That gives me

P_wave ~ P_tidal / N ~ P_tidal / ((120 hours)/(10 seconds))

which leads to the identical result of H ~ 10 m, as it turns out. Same effects as before, except that now my hypothetical forest and cliff are being demolished gradually by a succession of thousands of ordinary (though large) waves instead of suddenly by a solitary wave travelling at several tenths of a Mach.

To be continued; comments would be appreciated.

mfb

Tides are not surface waves: the whole water moves, which gives much more flow per energy.

Another issue: With a perihelion of ~15 million km, you will see significant effects from the double-star system, giving perturbations of the orbit.
If the stars have a separation of just ~1 million km (too low, the outer parts of the orange dwarf would just fly away due to tidal effects), the gravitational acceleration at perihelion varies by O(1%). Even if first and second order effects cancel, the orbit would change completely within millions of years.

onomatomanic

Yes, I'm aware that the comparison is a bad one. But since I've not yet succeeded in finding anyone who's both able and willing to tackle this in a more appropriate way, it's the best I have, for the time being.

Is there any way (short of a simulation) of figuring out in what way it would change? Everything is prograde and the binary's period is short compared to the progress of the planet along its orbit even around perihelion (obviously), if that helps.

The parameters of the binary system were chosen such that there is mass flow from the main-sequence star towards the compact object. That's how this "flying away due to tidal effects" usually materializes in this case, isn't it? I'm positing that the flow rate is low enough for the main sequence star to survive for billions of years. If that latter idea is unrealistic (is it?) that would be one of those breaks with reality that I can live with.

mfb

Hmm... simulation? There are some weird orbits, see Kepler 16b for an example (simulation). But the general case is chaotic, so it is hard to predict anything. The elliptical orbit will not help in that respect.

It would reach the compact object in some way.

Good idea, but it requires a minimal separation. No idea how much, but I would expect a few times the stellar radius. I would expect that life evolved after the white dwarf formed (otherwise it would have needed a different orbit before, to survive the end of the star), therefore the system has to be quite old.

onomatomanic

I was hoping that this might be one of those cases in which the three-body problem has an approximate analytical solution, as it is in the Sun-Earth-Moon case. But I guess that even if my case meets the criteria for such a solution to apply in the short term, which I'm firmly assuming it does, what you're talking about is the cumulative effect of the perturbation exerted on one of the components (the planet) by the other (the binary), which is by definition not part of the approximate solution, one would presume. Bah.

Anyway, I set up the simulator you linked to for my system. It seems that, for a circular planetary orbit about this kind of binary, the lower limit for short-term stability lies somewhere between 2 times (play) and 2.5 times (play) the separation of the stars. You might have to fiddle with the delay parameter, if that one runs too fast to see anything. According to the wikipedia article, the commonly accepted value for long-term stability is about thrice that.

If one now increases the initial velocity in the two cases to just short of escape velocity, resulting in highly eccentric orbits, 2 times stellar separation behaves pretty much as before (play), whereas 2.5 times seems more or less stable but shows definite changes over the course of many orbits (play).

As you predicted, that case turns out to be quite complex. I can make out at least three components to those changes: In the short term, there is an erratic fluctuation in the aphelion distance. Presumably, that depends on the phase of the binary during perihelion passage, which can slightly speed up or slow down the planet. In the middle term, the axis of the ellipse rotates in a prograde direction. That seems intuitively right for this solar system, somehow. In the long term, the planet disappears. That is, I left the simulation running for in the background for ten minutes after watching it for a while, and when I looked at it again, no more planet. I couldn't tell if it fell into one of the stars or was flung out of the system, but the latter seems far more likely. I guess the lesson here, if any, is that the perturbation doesn't act to eventually smooth out the orbit more and more, which is what I had sort of expected.

I've now increased the perihelion separation to 3 times stellar separation and am going to let that run for at least an hour. So far, so uneventful.

I simply chose that separation at which the gravity from either star cancels out at the surface of the orange dwarf. M1/R1^2 == M2/(d12-R1)^2. Is that too simplistic? Obviously, the star wouldn't be spherical any more but tear-drop shaped due to a combination of centrifugal and tidal deformation, so this R1 isn't quite the Main-Sequence-model radius for the given mass, but that should be a minor correction. Less than a factor of 2, surely.

I'm not planning on making the backstory of the solar system part of the plot in any way, so I don't really need to come up with one, but I have of course thought about it.

As you say, life couldn't be expected to survive the transition from main-sequence star to white dwarf. However, I've been assuming that on this orbit, the planet itself couldn't have survived that, either, considering that our Sun is predicted to swallow its inner planets during its expansion, AFAIK. Something would have to have changed the orbit later on in any case, therefore. I've been playing with the idea that it may already have been in a circular orbit in the then-habitable zone before the transition, and that life did arise then as well, only to be completely sterilized by the transition itself. Some time after the transition, once the white dwarf had time to cool down a bit to lower its hard radiation output to acceptable levels, Something Happened which put the planet into its current orbit which just so happens to once again make it habitable, and life independently begins and evolves a second time. Ideally, that same Something should also explain why the planet now has water and an atmosphere despite having been thoroughly charbroiled.

The payoff of this idea would be that there could theoretically be fossils from the earlier iteration, which would look (and be) a lot more alien to my Altlings (humanoid protagonists) than, say, dinosaurs look to us. But I haven't looked into how practical it is to expect fossils to survive both the charbroiling and the several billion years since then. So, for now, I'd classify this as idle speculation.

mfb

In the simulation, all objects are point-like, so it was flung out with an unknown minimal distance to the stars.
I am watching it while I am typing this post: The second time the aphelion is "at 8 o'clock" the orbit is very unstable and highly eccentric. It gets a bit more regular and then irregular again, and after one of those orbits it leaves the screen and needs >30s to come back (aphelion at 5 o'clock). Afterwards, the eccentricity is reduced again (aphelion moved to 10 o'clock now, and is just ~3 times the perihelion), but I think it will leave the system after a while in a similar way.

1 hour is ~1000 years?


In this case, the pressure gradient would be 0 (or even negative, to account for radiation pressure), this cannot be the surface of a star. Hmm... using the average density of orange dwarfs and the requirement that the whole star (even with deformation of ~factor 2) is outside the roche limit for liquid objects could give some approximation.

onomatomanic

Ah, yes, that makes sense. When I first played with the simulator and accidentally set up configurations which weren't even remotely stable, there were several occasions on which it looked like a planet had fallen into a star, so I assumed that was what had happened. But if the planet did a slingshot and then immediately went off-screen never to return, the two cases would be impossible to tell apart, of course.

I was thinking I could extrapolate from the circular-orbit case to this one. There, the value for short-term stability was a third of the value quoted for long-term stability in the article. Vice versa, once I find a minimal short-term-stable relative perihelion separation for elliptical orbits, what's to stop me from assuming that three times that value is long-term stable?

So, as it turns out, 3 times stellar separation destabilizes after about half an hour. 3.5 times, though, no longer exhibits the erratic fluctuations to any noticeable degree. The "precession", if that term applies here, is still present, but that needn't be a problem per se. The state of the system after an hour is indistinguishable from the initial state, so I'd say this qualifies as short-term stable now. Which means that my case, with about an order of magnitude between stellar and perihelion distances, should be fine. Borderline perhaps, but since my threshold is plausibility rather than realism, that's good enough.

Just to see what would happen to the 3.5 times stellar separation case after many more orbits, I tried reducing the number of "substeps" from 60 to 10 so it'd run faster, which it did. Unfortunately, that resolution turns out to be insufficient to model the stellar orbits sufficiently well, as those noticeably shrunk within the first ten minutes. The discrepancy between the timescales involved is just too great for this simulator, it seems.

Yeah, but that's sort of the point, isn't it? The criterion only applies to a small region directly "underneath" the compact object, and in that region, there is indeed no surface because the body of the star seamlessly turns into the streamer which connects the star with the compact object or the accretion disk around it or wherever else the matter flow ends up. If I'm missing something fundamental here, just point me in the right direction and I'll read up on it.

mfb

Hmm well, a constant stream does not look very stable within the timescale of several billion years.
Ok, I think this can be done with a bit hand-waving. You need this anyway, unless you find a plausible way how the system evolved. Some billion years to get enough heavy elements for the stars, some billion years to get a white dwarf, and then some billion years to get intelligent life could be a tight schedule. The stars needed more separation earlier in their life, too. A method to capture the white dwarf would be interesting, it would fix the timescale issue and could explain the elliptical orbit at the same time.

onomatomanic

^ Ohhh, that is perfect. Howdidntithinkofthat-itseemssoobviousinretrospect perfect, because it matches the Altling's creation myth which I came up with a while ago. The myth holds that both their planet and the "ear", which is what they colloquially call the primary, are part of the physical body of Eden Earthmistress (aka The Mother of Mothers, The Lady of The Land, The Weaver of Waters, or simply The Goddess), whereas the "eye", i.e. the secondary, is the only visible bit of Altar Allfather (aka The Sky-Father, The Lord of The Heavens, The Lord of Fire (I know, His stylings lack the Added Alliterative Appeal that Hers have - work in progress), or simply The God), the rest being hidden behind The Watery Veil (the blue sky, to us) which She wove for Him.

Anyway, according to the myth, Eden was alone and barren (in the land-sense as well as the woman-sense) in the beginning, until Altar met her, beheld her, loved her, and has been dancing circles around her ever since, which is of course why the suns rise and set each day. There's a lot more detail, needless to say, and some of that detail mysteriously (from the reader's perspective, that is) reflects physical and prehistorical truths about their world which the Altlings shouldn't have any knowledge of, given their level of scientific progress.

So, having the white dwarf be the late arrival doesn't just elegantly get around some of the practical problems we're discussing, it also ties beautifully into the cultural backstory that's already in place. As to capture method, I'm thinking the orange dwarf was already part of a binary, and the rogue white dwarf ended up transferring most of its kinetic overhead to that third star, thus effectively switching roles with it. That should work, no? (ETA: Yes, it should.)

mfb

Nice simulation of the stars.
Something like this could give the eccentric orbit of the planet, where the parameters would need some tweaking.

onomatomanic

I call this one "The Beauty of Chaos".