The surface in Figure 27.3 is the Endurance’s gravitational energy plus its centrifugal energy plotted vertically, and location in Gargantua’s equatorial plane plotted horizontally. Wherever the surface dips downward, the Endurance’s gravitational plus centrifugal energy decreases, so its radial kinetic energy must increase (since the total energy is unchanged); its radial motion must speed up. This is precisely what happens in our intuitive, volcano analogy.
Outside the moat of Figure 27.3, the surface’s height is controlled by the Endurance’s negative gravitational energy (see the “gravitational energy” label on the figure). By comparison, there the positive centrifugal energy is unimportant. On the outer edge of the volcano, by contrast, the height is controlled by the rising centrifugal energy, which has come to dominate over the gravitational energy. On the inside of the volcano, near Gargantua’s horizon, the gravitational energy has grown hugely negative and overwhelms the centrifugal energy, so the surface plunges downward (Figure 27.5). The critical orbit is on the volcano’s rim.
The Critical Orbit: Balance of Centrifugal and Gravitational Forces
Upon reaching the volcano’s rim, the Endurance, ideally, would travel around and around it, at constant speed. Because it moves neither inward nor outward, the inward pull of gravity on the rim must precisely be counterbalanced by the outward centrifugal force that arises from the ship’s fast circumferential motion.
This indeed is the case, as shown in Figure 27.6—an analog of the force balance plot for Miller’s planet (Figure 17.2). At the Endurance’s critical orbit, the red curve (the inward gravitational pull on the Endurance) and the blue curve (the outward centrifugal force) cross, so the two forces are in balance.
However, the balance is unstable, as our volcano-rim analogy suggests.[50] If the Endurance is randomly pushed inward just a bit, then gravity overwhelms the centrifugal force (the red curve rises above the blue curve), so the Endurance is pulled on inward toward Gargantua’s horizon. If the Endurance is pushed outward just a bit, then the centrifugal force wins the battle with gravity (the blue curve is above the red curve), so the Endurance is pushed on outward, escaping Gargantua’s tight grip.
By contrast (as we saw in Chapter 17), on the orbit of Miller’s planet, the balance between the gravitational and centrifugal forces is stable.
Disaster on the Rim: Ejection of TARS and Cooper
In my science interpretation of the movie, the volcano’s rim is very narrow, so the critical orbit on the rim is exceedingly unstable. Tiny errors in navigation will send the Endurance careening down toward Gargantua (down into the volcano) or away from Gargantua (down toward the moat).
Errors are inevitable, so the Endurance’s course must be corrected, continually, by a well-designed feedback system, like an automobile’s cruise control but much better.
In my interpretation, the feedback system is not quite good enough and the Endurance winds up dangerously far down the inside lip of the volcano. The Endurance must use all the thrust at its disposal to climb back up to the critical orbit.
But this is too subtle and technical for action-packed scenes and a hugely diverse audience, so Christopher Nolan chose a simpler, more in-your-face approach. No mention of instability. No mention of feedback. The Endurance simply plunges too close to Gargantua, and Cooper responds with all the thrust he can muster to climb back out and escape Gargantua’s grip.
The result is the same: lander 1, piloted by TARS, and Ranger 2, piloted by Cooper, fire their rockets while attached to the Endurance, pushing the Endurance back out of Gargantua’s gravitational grip. Then, to get the last possible kick, explosive bolts blow the Endurance apart from lander 1 and Ranger 2. The lander and Ranger go plunging downward toward Gargantua, carrying TARS and Cooper with them, and the Endurance is saved (Figures 27.7 and 27.8).
In the movie, there is a tragic, parting conversation between Brand and Cooper. Brand doesn’t understand why Cooper and TARS must accompany the lander and Ranger into the black hole. Cooper gives her a rather lame though poetic excuse: “Newton’s third law. The only way humans have ever figured out for getting somewhere is to leave something behind.”
This surely is true. But the additional thrust on the Endurance, from Cooper and TARS accompanying the lander and Ranger into the hole, is awfully small. The greater truth, of course, is that Cooper wants to go into Gargantua. He hopes that he and TARS can learn the quantum gravity laws from a singularity inside Gargantua, and somehow transmit them back to Earth. It is his last, desperate hope for saving all of humanity.
The Endurance’s Launch Toward Edmunds’ Planet
The critical orbit is an ideal spot for Brand and the robot Case to launch the Endurance in any desired direction, in particular, toward Edmunds’ planet.
How do they control their launch direction? Because the critical orbit is so unstable, a small rocket blast is sufficient to send the Endurance off it. And if the blast is ignited at precisely the right location along the critical orbit and has precisely the right strength, it will send the Endurance in precisely the desired direction (Figure 27.9).
50
The agreement between our volcano-rim analogy and these force arguments is due to a key fact: The net force (gravitational plus centrifugal) on the