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We will ignore some of the more obvious issues with the movie and assume all other things are consistent to have fun with some of these questions.

Simple [hopefully] Pre-questions:

1) If the water is only a meter or so deep, then how can there be enough water to produce waves hundreds of meters in amplitude?
2) Are we to assume the source of the waves are tidal forces from the black hole nearby? If so, wouldn't this significantly alter the local gravity experienced by the crew?
3) Would the waves be more appropriately defined as gravity waves or shallow water waves?

In the case of shallow water waves, the phase speed is, assuming a wavelength ($\lambda$) much larger than the water depth ($h$), given by:
$$ \frac{\omega}{k} = \sqrt{g h} $$ We are told in the movie that $g_{w} = 1.3 \ g_{E}$, or ~12.71-12.78 $m \ s^{-2}$. If we assume a water depth of 1 meter, then the phase speed should have been ~3.6 $m \ s^{-1}$ (roughly 8 mph).

If we ignore surface tension for the moment and assume the waves were gravity waves, then their phase speed is given by:
$$ \frac{\omega}{k} = \sqrt{\frac{g}{k}} \sim \sqrt{\frac{g \ \lambda}{2 \ \pi}} $$ From my limited memory, I would estimate that the wavelength of these waves was ~100-1000 meters (let's make the numbers easy to deal with) and we already know the gravity, so we have phase speeds of ~14-45 $m \ s^{-1}$ (roughly 32-100 mph).

It's difficult to estimate speeds from a movie, but I am not sure if these results seem reasonable or not. The speeds are certainly more reasonable (i.e., they seem close to the actual movie speeds, I think) than I thought they would be prior to calculation, but the results bother me.

Intuitive Issue [and main question]

The soliton-like pulse of the waves in the movie makes me doubt both the movie and my estimates. The reason is that the phase speed of solitons depends upon their amplitude and FWHM. My intuition says that the amplitude of the waves alone should have resulted in much higher phase speeds than my estimates and the speeds shown in the movie.

Updates

I am not so much worried about the black hole or any direct general relativistic effect it might have on the planet. I am only interested in the waves on the planet.

Questions

  1. Can anyone suggest a possible explanation that might alleviate my concerns?
  2. The water is very shallow, as shown by the characters walking through it. So how can there be several hundred meter waves?
    • Is it that all the planet's water is coalesced into these wave-like distortions (i.e., Are these just extreme tides?)?
    • Are the waves actual a distortion of the planet's surface and the water is still only a meter or so deep?
      • [Just to be nit-picky] If the previous question is true, then how would such a world not have significant volcanic activity (e.g., see Jupiter's moon Io)?
  3. If the waves are entirely water-based (i.e., they are effectively extreme tides) then their amplitude is orders of magnitude larger than the water depth or $\delta \eta/\eta_{o} \gg 1$. Is this a wave or just an extreme tide?
    • If a wave, then:
      • would the propagation speed of such a wave(?) be dominated by tidal effects?
      • would it act like a soliton-like pulse once formed?
    • If a tide, then:
      • would there not be (extreme?) weather changes near these mounds of water?
honeste_vivere
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    Can we start a tag for Interstellar-related questions? I would have so many ;) – SuperCiocia Jan 04 '15 at 18:13
  • I would have, but I do not think I have enough reputation to do so. I also have several other questions, but this one seemed like an approachable one (I hope). – honeste_vivere Jan 04 '15 at 18:16
  • I was joking, it wouldn't be very professional... the next tag would be Inception... ;) – SuperCiocia Jan 04 '15 at 18:23
  • I don't think the planet/bh scenario is nearly as interesting as people think. For a stellar size bh the Roche limit of the planet is deep inside classical gravity territory. That's probably pretty much equivalent to a tidal locked planet around an ordinary star. I think the much more interesting cases are that of 1) Early moon, where a small moon is near its Roche limit orbiting the planet 2) The Melancholia scenario, where two planet size bodies nearly collide. That is when the crust really hits the fan! The effects shown in "Melancholia" were pretty close IMHO. – CuriousOne Jan 04 '15 at 18:23
  • @CuriousOne - It may not be interesting in that sense, but I wasn't really worried about that. I was curious about the dynamics of the water waves, how they form, and whether they were described appropriately. – honeste_vivere Jan 04 '15 at 18:28
  • @SuperCiocia - I was serious in so far as wanting to make it easier to ask GR specialists about some things in the movie. I took GR in graduate school, but classes are often over simplified versions of actual research and understanding. So I doubt my understanding of GR is deep enough to really tackle some of the things I have questions about (not the obvious dramatic liberties). – honeste_vivere Jan 04 '15 at 18:31
  • That's my point: if you are so far away that it's basically classical Newtonian gravity, then you are in a tidal lock, except that there is an ultrabright gamma-ray point source in the sky (you are looking at an object a few miles in size from a few million miles away, or so!). Needless to say... that thing would fry your oceans in no time. – CuriousOne Jan 04 '15 at 18:32
  • @CuriousOne - Oh, are you worried about my second question? If so, I think you answered it in your comments. So ignoring the BH, what would your intuition about the propagation of such waves be relative to what was shown in the movie? I am trying to determine whether my concerns are valid (i.e., the waves should have moved significantly faster). – honeste_vivere Jan 04 '15 at 18:39
  • I am basically just saying that the bh doesn't add anything to a tidal locked planet scenario. I think it probably takes a lot of good physics away because the more interesting things are happening when a smaller body is very close to the surface of a bigger one. The stellar bulge underneath a hot Jupiter is where the physical beef is, and the bh doesn't have that, at all. Just my two cents. – CuriousOne Jan 04 '15 at 18:42
  • 300km tidal bulge: http://www.skyandtelescope.com/astronomy-news/exoplanets/hot-jupiters-keep-stars-young/ – CuriousOne Jan 04 '15 at 18:45
  • @CuriousOne - Wow, that is certainly a significant tidal effect (waves destroying LEO spacecraft would definitely be an issue)... – honeste_vivere Jan 04 '15 at 18:48
  • NB: Your title doesn't seem to match the question (specifically the "source of water" bit). The Interstellar wiki on Planet Miller puts the amplitudes at about 1200 m & discusses more aspects of the planet. I suspect there is more in Thorne's book The Science of "Interstellar". – Kyle Kanos Oct 09 '15 at 12:49
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    Who says that tides aren't waves? See Does Earth really have two high-tide bulges on opposite sides? for more details, but even if, on Earth, you (i) took away the land, and (ii) made all oceans as deep as the deepest trench, you would still have significant wave effects coming from the fact that the water simply cannot respond fast enough to the changes in gravity caused by the Moon's tidal force. Moreover, once you get to amplitudes as high as the depth itself, there are strong nonlinear effects to be considered. – Emilio Pisanty Oct 09 '15 at 12:50
  • @KyleKanos - Ah, yes I see my amplitudes were off. My question stemmed from my memory after having seen the movie only once. The "source of water" bit comes from the interaction of their spacecraft with the wave. It did not seem like a floating craft, so I was a bit confused when they rode the wave to the top of the crest. I was also confused as to how one could get such large waves when the water is only ~1 m deep, thus why I chose those words. Do you have a better suggestion? – honeste_vivere Oct 09 '15 at 13:02
  • @honeste_vivere: I would have used Explaining the waves on the water planet in Interstellar or something similar. – Kyle Kanos Oct 09 '15 at 13:07
  • @EmilioPisanty - So I was trying to be careful because the definition of a wave is somewhat tenuous and difficult to constrain (I'm paraphrasing Whitham a bit here). For instance, in plasmas there are modes that are effectively purely growing modes, i.e., $\Re (\omega) = 0$ but $\Im (\omega) \neq 0$. In addition, part of the reason for my question is that nonlinear effects should be critical here but they did not seem to be. Meaning, a rogue wave for instance, of that amplitude on Earth would move much faster than the waves are depicted to move in the movie. – honeste_vivere Oct 09 '15 at 13:08
  • I agree, there's definitely lots of interesting hydrodynamics to explore to see how plausible the waves are. Some of it depends on canon, which is not very conclusive, but Thorne's The Science of "Interstellar" is as close to canon as you'll get on this, I think. (I.e.: you should look at it and update your post to include the conditions it gives for Miller's planet, in particular its surface gravity and the strength of the tidal forces. Keep your post self-contained to the physics and minimal speculation.) – Emilio Pisanty Oct 09 '15 at 13:19
  • I also take issue with questions of the form "how can there be enough water?". The water is obviously there, why does it need to 'come from somewhere'? The real question is what conditions would sustain the concentration of water into big waves in a shallow pool instead of having the waves dissipate into smaller waves in a deeper pool. – Emilio Pisanty Oct 09 '15 at 13:21
  • @EmilioPisanty - Generally, somewhat radical things are required for a water wave to have an amplitude that greatly exceeds the depth of the surrounding water (ignoring shallow water run-up effects like the jaws waves near Maui), which provoked my question. It's as if all the water is somehow isolated within the wave. I will, however, look into getting the book and reading about the planet. – honeste_vivere Oct 09 '15 at 16:58
  • @EmilioPisanty - Have book and currently at chapter 19. His chapter on the water planet was very hand wavy and not tremendously enlightening though. I will keep reading to the end and the post an answer... – honeste_vivere Jan 24 '16 at 15:31

2 Answers2

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The following interpretations are taken from Thorne [2014].

Chapter 17, entitled Miller's Planet, discusses the issue of the large waves on the water planet in the movie Interstellar. There Kip mentions that the waves are due to tidal bore waves with height of ~1.2 km. In the appendix entitled Some Technical Notes, Kip estimates the density of Miller's planet to be $\sim 10^{4} \ kg \ m^{-3}$. For comparison, Earth's density is $\sim 5.514 \times 10^{3} \ kg \ m^{-3}$. We are also told that the planet itself has ~130% of the gravitational acceleration of Earth. From this we can estimate the mass and radius of Miller's planet (ignoring tidal distortions to make things easy): $$ \begin{align} r_{M} & = \frac{3 g_{M}}{4 \pi \ G \ \rho_{M}} \tag{1a} \\ & = \frac{3.9 g_{E}}{4 \pi \ G \ \rho_{M}} \tag{1b} \\ r_{M} & \sim 4546-4572 \ km \tag{1c} \\ M_{M} & = \frac{ 9 g_{M}^{3} }{ \left( 4 \pi \ \rho_{M} \right)^{2} \ G^{3} \ \rho_{M}} \tag{2a} \\ & = \frac{ 19.773 g_{E}^{3} }{ \left( 4 \pi \ \rho_{M} \right)^{2} \ G^{3} \ \rho_{M}} \tag{2b} \\ M_{M} & \sim 3.936 \times 10^{24} - 4.002 \times 10^{24} \ kg \tag{2c} \end{align} $$

For reference, the Earth's mean equatorial radius is $\sim 6.3781366 \times 10^{3} \ km$ and the Earth's mass is $\sim 5.9722 \times 10^{24} \ kg$.

The water is very shallow, as shown by the characters walking through it. So how can there be several hundred meter waves?

Unfortunately, the answer is extremely boring. The planet is tidally locked with the nearby black hole and nearly all of the surface water of the planet is locked into two regions on opposite sides of the planet. The planet itself is shaped much like an American football rather than an oblate spheroid.

There is a slight problem with this interpretation, though. In the movie, the Ranger appears to float. Though I do not doubt that the vehicle is well sealed, I am curious if it could displace more water than its weight allowing it to float on the massive waves.

Is this a wave or just an extreme tide?

Just an extreme tide, and according to the wiki on this planet, they do not actually propagate, the planet rotates beneath you due to a slight difference in the planet's rotation rate and its orbital motion (i.e., the planet "rocks" back-and-forth during its orbit about the black hole).

would there not be (extreme?) weather changes near these mounds of water?

I would be very surprised if such large mounds of water were not surrounded by or at least affecting the nearby weather, much the same as mountains on Earth. However, this is starting to split hairs in an already speculative subject I guess.

Updated Thoughts

I updated the following computations for fun merely because I found them more interesting than the tidal bores.

Gravity Waves
If we assume that the wave height were the same as the wavelength and we assume these were gravity waves, then their phase speed would be ~49 m/s.

Shallow Water Waves
If we assume the wavelength is $\sim r_{M} \gg h$ (i.e., from Equation 1c), where we now assume $h$ ~ 1.2 km, then the phase speed would go to ~124 m/s.

References

  • Thorne, K. "The Science of Interstellar," W.W. Norton & Company, New York, NY, ISBN:978-0-393-35137-8, 2014.

Typos and/or mistakes in book

I only found a few typos/mistakes in the book, which are listed below:

  • Chapter 2
    • He confuses the north and south magnetic poles (i.e., the north magnetic pole is located near the south geographic pole, not the north).
    • He assigns the source of the aurora to protons. However, the aurora are due to energetic electrons exciting oxygen and nitrogen.
  • Chapter 7
    • He states that the Cassini spacecraft used "...Saturn's moon Io..." for a gravitational slingshot. However, Io is one of the four Galilean moons of Jupiter and Saturn is the planet to which Cassini was headed.

I consider these fairly minor and honest mistakes, but worth taking note of...

honeste_vivere
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The issue with massive waves on a 1 meter deep ocean is that the waves cannot propagate fast enough on a planetary size object. We get fast moving shallow tsunami waves in the open ocean over a thousand meters deep. The tsunami piles up when the wave slows down due to contact with a shallow shoreline. Hundred meter high waves could never propagate fast enough in a one meter deep ocean to pile up. The same for a tidal bulge, water could not flow fast enough around a planetary distance to pile up in a one meter deep ocean.

What might be possible is that tidal effects on the land would be large enough to slosh waves of considerable height, especially if there were a sypathetic frequency. On Earth the land tides are a few inches, close to a black hole there might be land tides of over a meter. Of course, this creates problems of its own - since the energy absorbed from such huge land motions would remelt the planet.

M Willey
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