The Science of Interstellar

The equations that I gave to Oliver James at Double Negative, for the orbital motion of light rays around Gargantua, are a variant of those in Appendix A of Levin and Perez-Giz (2008). Our equations for the evolution of bundles of rays are a variant of those in Pineult and Roeder (1977a) and Pineult and Roder (1977b). In several papers that we’ll make available at http://arxiv.org/find/gr-qc, Paul Franklin’s team and I give the specific forms of our equations and discuss details of their implementation and the simulations that resulted.

Chapter 12. Gasping for Oxygen

Here are the calculations that underlie my statements in Chapter 13. They are a nice example of how a scientist makes estimates. These numbers are very approximate; I quote them accurate to only one digit.

The mass of the Earth’s atmosphere is 5 × 10¹⁸ kilograms, of which about 80 percent is nitrogen and 20 percent is molecular oxygen, O₂—that is, 1 × 10¹⁸ kilograms of O₂. The amount of carbon in undecayed plant life (called “organic carbon” by geophysicists) is about 3 × 10¹⁵ kilograms, with roughly half in the oceans’ surface layers and half on land (Table 1 of Hedges and Keil [1995]). Both forms get oxidized (converted to CO₂) in about thirty years on average. Since CO₂ has two oxygen atoms (that come from the atmosphere) and just one carbon atom, and the mass of each oxygen atom is 16/12 that of a carbon atom, the oxidization of all this carbon, after all plants die, would eat up 2 × 16/12 × (3 × 10¹⁵ kilograms) = 1 × 10¹⁶ kilograms of O₂, which is 1 percent of the atmosphere’s oxygen.

For evidence of sudden overturns of the Earth’s oceans and the theory of how they might be produced, see Adkins, Ingersoll, and Pasquero (2005). The standard estimate of the amount of organic carbon in sediments on the ocean bottoms that might be brought to the surface by such an overturn focuses on an upper sedimentary layer that is mixed by ocean currents and animal activity. This mixed layer’s carbon content is the product of an estimated rate of deposit of carbon into the sediments (about 10¹¹ kilograms per year) and the average time it takes for its carbon to be oxidized by oxygen from ocean water (1000 years), giving 1.5 × 10¹⁴ kilograms, one-twentieth of that on land and in ocean surface layers (Emerson and Hedges 1988, Hedges and Keil 1995). However: (i) The estimated deposition rate could be wrong by a huge amount; for example, Baumgart et al. (2009), relying on extensive measurements, estimate a deposition rate in the Indian Ocean off Java and Sumatra that is uncertain by a factor of fifty and, extrapolated to the whole ocean could give as much as 3 × 10¹⁵ kilograms of carbon in the mixed layer (the same as on land and in the ocean’s surface layers). (ii) A substantial fraction of the deposited carbon could sink into a lower layer of sediment that does not get mixed into contact with seawater and oxidized except possibly during sudden ocean overturns. The last overturn is thought to have been during the most recent ice age, about 20,000 years ago—twenty times longer than the oxidation time in the mixed layer. So the unmixed layer could have twenty times more organic carbon than the mixed layer, and as much as twenty times that on land and in the ocean’s surface. If brought to the ocean surface by a new overturn and there oxidized, this is nearly enough to leave everyone gasping for oxygen and dying of CO₂ poisoning; see the end of Chapter 12. Thus such a scenario is conceivable, though highly unlikely.

Chapter 15. Visualizing Interstellar’s Wormhole

Christopher Nolan chose several kilometers for the diameter of Interstellar’s wormhole. The wormhole’s angular diameter as seen from Earth, in radians, is this diameter divided by its distance from Earth, which is about 9 astronomical units or 1.4 × 10⁹ kilometers (the radius of Saturn’s orbit). Therefore, the wormhole’s angular diameter is about (2 kilometers)/(1.4 × 10⁹ kilometers) = 1.4 × 10^–9 radians, which is 0.0003 arc-seconds. Radio telescopes routinely achieve this angular resolution using transworld interferometry. Optical telescopes on the ground using a technique called “adaptive optics,” and the Hubble space telescope in space, achieve angular resolutions a hundred times worse than this in 2014. Interferometry between twin Keck telescopes in Hawaii in 2014 can achieve a resolution ten times worse than the wormhole’s angular diameter, and it is very plausible that in the era of Interstellar optical interferometry between more widely spaced optical telescopes will make possible resolutions better than the wormhole’s 0.0003 arc-seconds.

Chapter 17. Miller’s Planet