What would tides look like on a planet with two moons?

[JoeSchmuckatelli]

Picture a wet terrestrial planet much like ours but with two moons.  Individually, each moon is less massive than ours, but together weigh more.  (say 3/4 lunar mass and 1/2 lm. Guessing 1/2 lm is sufficient for a moon to be spherical?) Their orbital paths don't have to be perfectly aligned but should be in the same plane. 

What would the tides look like on such a world? 

Sadly I don't have the maths/physics background to try to figure out where these moons might orbit (distance, etc) to be stable - and then further figure out what the frequency might be relative to the sun to estimate the tides... But I am interested. 

Anyone willing to help me with this? 

... 

I can predict cycles of highest tides based on both moons being aligned with the sun and middling tides when perpendicular and something different when neither are in line with or perpendicular to the star. 

What I don't know is what periods this might look like (we have 2-week cycles with one moon on a monthly orbit) - because I don't have the skill to figure out what orbits might be stable given the criteria I set up (two moons with a collective mass higher than our own ('moon' used on purpose)). 

I also can't figure out what tides might look like with both moons 'new' and whether the tides might differ if both were full, or one new and one full.  I suspect that this matters, but I do not know how much. 

Anyway - I know this is probably a big ask, but I would appreciate anyone willing to help me figure out what this might look like. 

Edited July 16, 2021 by JoeSchmuckatelli

[tomf]

Our tides are pretty much the sum of two sine waves, one with a period of a day from earth's rotation relative to the sun, and the other of slightly longer period due to the earth's rotation relative to the moon.

You do get high tides on both the side near the moon and furthest from it so the new moon is basically equivalent to the full moon.

When these are in phase withe the earth moon sun forming a straight line (either way round) they add and you get big Spring tides, when they are at 90 degrees they are out if phase and you get smaller neap tides.

The moon sine wave has bigger amplitude so the highs and lows follow the moon's cycle of of 25 hours or so.

With two moons you are going to be adding three waves. The biggest tides will indeed come when everything is in a straight line.

The smallest tides will probably come when smaller moon and sun are at 90 degrees from the bigger moon.

Jupiter seems to manage well with its moons in a 4:2:1 resonance so your moons could be something like that.

Actual tides are more complicated and depend on coastline shape and ocean basin shape as well as the astronomy.

Edited July 16, 2021 by tomf

[JoeSchmuckatelli]

56 minutes ago, tomf said:

The smallest tides will probably come when smaller moon and sun are at 90 degrees from the bigger moon.

Huh... I figured that the larger moon would have to be at 90 and the smaller at something like 45 to the sun for the smallest tides.  (both at 90 keeps the smaller aligned with the sun)

 

58 minutes ago, tomf said:

Actual tides are more complicated and depend on coastline shape and ocean basin shape as well as the astronomy.

This is one of those things that's easy to forget - until you look into the Big Wave competitions in Portugal, as well as some of the larger river tides (can't think of the names off the top, but there's a well known one in Canada, an impressive tide in China and IIRC something really interesting in Scotland.

 

1 hour ago, tomf said:

Jupiter seems to manage well with its moons in a 4:2:1 resonance so your moons could be something like that.

One of the things I was trying to picture was whether something like a 3:2 resonance would work... and then if so, how often you'd actually get all three tide-ing (??) bodies aligned.  If the larger moon were on a similar 1-month (ish) orbit, how do I figure out how often in a year one could expect all three to be aligned to tell prospective surfers and fishermen when to expect Spring tides?

 

(oh - and thanks for the response!)

[StrandedonEarth]

1 hour ago, JoeSchmuckatelli said:

but there's a well known one in Canada

Bay of Fundy, between the provinces of Nova Scotia and New Brunswick:

Spoiler

 

[NFUN]

2 hours ago, JoeSchmuckatelli said:

One of the things I was trying to picture was whether something like a 3:2 resonance would work... and then if so, how often you'd actually get all three tide-ing (??) bodies aligned.  If the larger moon were on a similar 1-month (ish) orbit, how do I figure out how often in a year one could expect all three to be aligned to tell prospective surfers and fishermen when to expect Spring tides?

it's just the least common multiple of the two orbital periods, assuming this is in reference to Earth, no the sidereal period (in which case, add a few days)

[wumpus]

The Earth's moon is rather large compared to the size of Earth.  Adding a second Luna sized moon to the Earth would probably require something like the Jovian system or more likely one of the sitting in a Lagrange point.

Beyond that, it looks like everything was covered in Tomf's post.

[sevenperforce]

Typically, 2:3 or 1:2 or 3:4 orbital resonance is necessary to make sure the moons remain stable. Remember that the strength of a tide decreases with the cube of the distance but increases linearly with the mass of the satellite, so if you had a moon 1/9 the mass of ours that was just 1/3 the distance, it would produce tides of the same magnitude.

Next, keep in mind that the square of the period of any satellite is proportional to the cube of the radius. So if you want a 2:3 resonance, you need the nearer moon at an orbital radius that’s about 76% of the radius of the more distant moon (because ((2/3)^2)^(1/3) = 0.7631). If you want a 1:2 resonance then the nearer moon orbits at 63% the radius of the distant moon; if you want a 3:4 resonance then it’s 82.5%. 

So...because of the way the cubes and squares interact, the tidal effect of any two moons in orbital resonance is proportional to the square of that orbital resonance. With a 1:2 resonance, the farther body will have 1/4 the tidal impact of the nearer body. With a 2:3 resonance, the farther body will have 4/9 the tidal impact of the nearer body. With a 3:4 resonance, the farther body will have 9/16 the tidal impact of the nearer body.

This gives you a good starting point. If you wanted a 3:4 resonance and you wanted both moons to produce tides of the same size, you’d want the farther body to be 16/9 the mass of the nearer body.

With our single moon, the tidal cycle is forced by a combination of the moon’s orbital eccentricity and the [smaller but still significant] solar tide. But with multiple moons, the resonances are going to force near-circularity and wash out any eccentricity effects. Comparing only the two moons, you get the most aggressive tides when the two moons line up and the mildest tides when they are opposite. 

[kerbiloid]

Tide height ~= 4.6 * H earth * (M tider / M tidee) * (R tidee / R earth) ⁴ * (Distance / 1 mln km)^-3

 

H earth = tide height in the open deep ocean on the Earth ~= 0.3 m

M tider/tidee = mass of the body causing the tices / the body on which is the tide

R tidee = radius of the body on which is the tide

R earth = Earth radius

Distance = distance between the boides

***

tider = Moon
tidee = Earth
~ 30 cm.

***

In the OP post the radius and distance of both bodies is required.

[tomf]

This graph shows the theoretical tides on a planet with two moons in a 3/2 resonance with periods of 20 and 30 days

https://www.desmos.com/calculator/qa050dsx5t

The red line is the influence of the inner moon with a 20 day period, the blue is the outer noon (30 day peiod period, 4/9 the effect strength) and the green is the sun (x is in days)

the purple is the sum of the effects of all three. As you can see the interval between high and low tides pretty much follows the strongest signal, but the spring neap cycle is more complicated.

[JoeSchmuckatelli]

18 hours ago, tomf said:

This graph shows the theoretical tides on a planet with two moons in a 3/2 resonance with periods of 20 and 30 days

https://www.desmos.com/calculator/qa050dsx5t

The red line is the influence of the inner moon with a 20 day period, the blue is the outer noon (30 day peiod period, 4/9 the effect strength) and the green is the sun (x is in days)

the purple is the sum of the effects of all three. As you can see the interval between high and low tides pretty much follows the strongest signal, but the spring neap cycle is more complicated.

I've really enjoyed perusing the graph you presented!  Interesting to see that there are periodical, off-beat pulses here and there: i.e. the tides at and around the lows at 500 and 1120 (x) are quite distinct from one another.

I can only imagine that such a planet would experience some very interesting coastal weathering and have periods of quite variable pelagic navigation (sometimes open water - other times vicious shallows).  3890 shows a single  big pulse in the high (above 1.5) where most other parts of the graph get a double high that barely reaches 1.5).  With the lowest lows looking to be ~ 0.3, the highest highs (globally) are over 5 times that of the lowest lows - and that's before you take into account shorelines or sea floor.  Probably be an interesting place to visit: heck of a place for a surfing competition!

There's also a 10 month cycle between the highest Spring tides - which would probably have some form of cultural significance to any intelligent life that evolved there (or lived there long enough).

 

Thanks very much for the time you (and others) took to answer this for me!

Edited July 18, 2021 by JoeSchmuckatelli

[JoeSchmuckatelli]

On 7/16/2021 at 8:43 PM, sevenperforce said:

This gives you a good starting point. If you wanted a 3:4 resonance and you wanted both moons to produce tides of the same size, you’d want the farther body to be 16/9 the mass of the nearer body

I had not thought about driving tides of the same magnitude via having the smaller moon inward of the larger - or tweaking mass like that.  I'm guessing what @tomf described is exactly what I was thinking: a larger inner moon and a smaller outer.

Without knowing - is it a fair guess/prediction that such an arrangement might have lower overall amplitudes but sharper delineations?

 

[YNM]

On 7/17/2021 at 7:43 AM, sevenperforce said:

Comparing only the two moons, you get the most aggressive tides when the two moons line up and the mildest tides when they are opposite. 

90 degrees maybe ? Opposite works precisely the same as when it lines up to one side.

Kind of wondering if calculating / figuring out where the bulge will point out might be a better way to tell when the tides are going to be highest or lowest. Also even on Earth there's technically 2 different tides - the tide of the solid body and the tide of the liquid ocean which does not happen at the same time.

Edited July 18, 2021 by YNM

[JoeSchmuckatelli]

10 hours ago, YNM said:

90 degrees maybe ? Opposite works precisely the same as when it lines up to one side.

Kind of wondering if calculating / figuring out where the bulge will point out might be a better way to tell when the tides are going to be highest or lowest. Also even on Earth there's technically 2 different tides - the tide of the solid body and the tide of the liquid ocean which does not happen at the same time.

You know - you bring up another something I've wondered about... but that our planetary scientists seemingly cannot resolve.

However, first to your point about the liquid ocean tides - they do indeed lag a bit behind the position of the moon... the explanation I've received is that (effectively) friction causes the slight lag between the height of the bulge (tide) and position of the moon.  Moon pulls everything up towards it, and it takes time for it to spill towards the pull, plus I'm guessing the passing of those inconvenient land masses / continents contributes somewhat to the delay in the oceanic tides.

But as far as terrestrial tides: we have planetary scientists saying that tides affecting Enceladus and other moons are driving internal convection which in turn contributes to cryo-vulcanism & etc.  I've yet to see anything definitive that describes the relationship between Lunar gravity and terrestrial vulcanism or continental drift / earthquakes.  Logically, if the Moon is massive enough to deform the planet, it has to have some effect on those continents and volcanoes... but I don't think anyone has yet figured out what that effect is.  Complicating matters is that Venus has no moon, and yet experiences vulcanism... which makes me wonder - how?

Grin - Science is fun.  Lots left to learn!

[RCgothic]

Now do multiple moons with eccentric inclined orbits. ;-)

[sevenperforce]

17 hours ago, JoeSchmuckatelli said:

I had not thought about driving tides of the same magnitude via having the smaller moon inward of the larger - or tweaking mass like that.  I'm guessing what @tomf described is exactly what I was thinking: a larger inner moon and a smaller outer.

Yeah I think we assume that you'd have a larger inner moon and a smaller outer -- probably in part because of KSP -- but moon formation doesn't necessarily operate like that at all. The Galilean moons go small-smallest-biggest-big in order from inside to outside. My intuition is that having a larger outer moon will help force a resonance but I don't know that for sure.

17 hours ago, JoeSchmuckatelli said:

Without knowing - is it a fair guess/prediction that such an arrangement might have lower overall amplitudes but sharper delineations?

I think you would have much higher amplitudes because the tides are similar in overall size.

13 hours ago, YNM said:

90 degrees maybe ? Opposite works precisely the same as when it lines up to one side.

Oh yeah, you're absolutely correct. I had that twisted. 

2 hours ago, JoeSchmuckatelli said:

But as far as terrestrial tides: we have planetary scientists saying that tides affecting Enceladus and other moons are driving internal convection which in turn contributes to cryo-vulcanism & etc.  I've yet to see anything definitive that describes the relationship between Lunar gravity and terrestrial vulcanism or continental drift / earthquakes.  Logically, if the Moon is massive enough to deform the planet, it has to have some effect on those continents and volcanoes... but I don't think anyone has yet figured out what that effect is.  Complicating matters is that Venus has no moon, and yet experiences vulcanism... which makes me wonder - how?

Jupiter has such absolutely massive gravity that its tides are strong enough to keep Io's mantle molten and to keep the subsurface ocean of Europa from freezing. Saturn doesn't have as much gravity as Jupiter, of course, but it is enough to drive cryovolcanoes on Enceladus.

But the amount of internal heating produced on Earth by our moon is comparatively negligible. The volcanoes on Earth and Venus are driven by internal heat from radioactive decay, since both Earth and Venus are massive enough to have accumulated quite a lot of radioactive material during solar system formation. Mars once had volcanoes, but it wasn't as massive and so what internal heat it used to have from radioactivity has long since cooled.

7 minutes ago, RCgothic said:

Now do multiple moons with eccentric inclined orbits. ;-)

You don't have to 'cause they would eject each other.

[kerbiloid]

A five-moon tide is even more weird.

Beware. Hide. Run.

https://www.goodreads.com/book/show/468120.When_the_Five_Moons_Rise

Spoiler

 

[JoeSchmuckatelli]

1 hour ago, sevenperforce said:

the amount of internal heating produced on Earth by our moon is comparatively negligible.

Huh.  

I pick through articles like this (mind you, grasping only generally the detail they try to explain) and I see that there is some speculation that lunar tides can affect already pressurized volcanic systems:  Effects of tidal stresses on volcanic activity at Mount Etna, Italy - Sottili - 2007 - Geophysical Research Letters - Wiley Online Library

1 hour ago, sevenperforce said:

volcanoes on Earth and Venus are driven by internal heat from radioactive decay, since both Earth and Venus are massive enough to have accumulated quite a lot of radioactive material during solar system formation. Mars once had volcanoes, but it wasn't as massive and so what internal heat it used to have from radioactivity has long since cooled

This comports with some of what I've read about the mantle convection* (e.g. ms3.dvi (yale.edu)) - but its interesting to me that they simply do not even mention tidal effects.  I would think that with an essentially fluid body (at scale) with a semi-rigid crust, that significant tidal pressure from the moon might at least contribute to 'stirring the pot'.  This article, OTOH talks about how Earth tides keep the Moon's core warmed up: Still hot inside the Moon: Tidal heating in the deepest part of the lunar mantle | NAOJ: National Astronomical Observatory of Japan - English  ...and while it does not mention the Earth being warmed by the Moon (aside from some weak implication for further research at the end) - it does make me wonder whether the Moon's influence does contribute to keeping our core hotter than it would be without such a large moon.

The thing I find interesting is the absence, at least from my reading, of papers saying

- 'nope, the Moon does not heat the Earth's core'

or

- 'Earth mantle convection moderated by Lunar tides'

...makes me wonder whether the big convection papers are even considering the possibility, even if it were only to discount it.

 

 

*Makes me also wonder if the Earth's heat is almost solely due to radioactive decay and internal convection, whether the planet will survive 'alive' long enough for the sun to reach its Red Giant stage - or if it will be an inert rock ball like Mars well before then.

Edited July 18, 2021 by JoeSchmuckatelli

[sevenperforce]

29 minutes ago, JoeSchmuckatelli said:

The thing I find interesting is the absence, at least from my reading, of papers saying

- 'nope, the Moon does not heat the Earth's core'

or

- 'Earth mantle convection moderated by Lunar tides'

...makes me wonder whether the big convection papers are even considering the possibility, even if it were only to discount it.

The moon absolutely does contribute to heating in the Earth's mantle (physics says that frictional heating has to go somewhere), but it is comparatively negligible when you put it next to the degree of tidal heating that the moons of Saturn and Jupiter experience.

One reason that it's not a 1-to-1 comparison is that the mechanism for tides is different with the gas-giant moons than it is for Earth. Lunar tides on Earth are caused primarily by Earth's rotation with respect to the moon's gravity. As the Earth rotates, the tidal bulge remains pointed toward the moon, so from Earth's perspective the bulge is traveling around the Earth. In contrast, our moon (as well as all the large moons of Jupiter and Saturn) are tidally locked, so their bulge doesn't rotate from the perspective of the surface. Instead, the bulge gets larger and smaller between periapsis and apoapsis, and that squeezing-and-stretching cycle is what causes frictional heating. A tidally-locked moon in a perfectly-circular orbit would have a tidal bulge but it would never change and so it wouldn't have any tidal heating.

41 minutes ago, JoeSchmuckatelli said:

Makes me also wonder if the Earth's heat is almost solely due to radioactive decay and internal convection, whether the planet will survive 'alive' long enough for the sun to reach its Red Giant stage - or if it will be an inert rock ball like Mars well before then.

The power output of Earth's core and mantle from radioactive decay is approximately 44 trillion watts. The total power output of all ocean tides is about 3 trillion watts, and only a fraction of that is dissipated as heat. The oceans experience much more significant lunar tides than the Earth's mantle, but of course Earth's mantle is a lot more "there" and so I'm not sure whether internal lunar tides would have greater power than ocean lunar tides.

[JoeSchmuckatelli]

37 minutes ago, sevenperforce said:

the bulge gets larger and smaller between periapsis and apoapsis, and that squeezing-and-stretching cycle is what causes frictional heating.

Elegant answer - thanks.

 

37 minutes ago, sevenperforce said:

The power output of Earth's core and mantle from radioactive decay is approximately 44 trillion watts

My 'Makes me wonder' posit was more about general planetary cooling - yes, an aside from what we've been discussing - meaning, if internal radiation is the cause of our heat, and lunar tides are so negligible, what does Earth's 'cooling to a hard rock' timeline look like compared to the time it will take for the Sun to reach Red Giant stage?  I.E. whether Earth's core will remain hot and relatively liquid / active (meaning it could conceivably retain an atmosphere, and life)  for the next 5.4 billion years, or if it will have cooled to a rock before the Sun morphs into a Red Giant

[sevenperforce]

1 hour ago, JoeSchmuckatelli said:

what does Earth's 'cooling to a hard rock' timeline look like compared to the time it will take for the Sun to reach Red Giant stage?  I.E. whether Earth's core will remain hot and relatively liquid / active (meaning it could conceivably retain an atmosphere, and life)  for the next 5.4 billion years, or if it will have cooled to a rock before the Sun morphs into a Red Giant

It's estimated that Earth's core would take 91 billion years to cool to the "hard rock" stage.

[kerbiloid]

Though, it's estimated that it needs ~1.5 bln to get fully differentiated and stop the geological processes and cancel the magnitosphere.

The radioactive input is just about 15% or so.

And keeps decreasing due to decay.

Also the Moon is almost at the end of its presence in the Earth life, as its orbit inclination has already decreased from original 23.5° (the Earth equator plane and at the same time the original orbital plane of the Moon created on the (impact / close interaction) event) down to the current 5°, matching the ecliptic plane.

So, it required 4.5 bln to pass fromthe "close sat of the Earth" to "Earth companion", at it needs to rotate to the orbit plane for 1/4 of that to get to the ecliptic plane, and become a "solar sat rented by Earth".

I.e. 2 bln years later there will be a dead, post-Earth, New Venus with no tides on the hot ground from the Sun's Moon.

Edited July 19, 2021 by kerbiloid