by Thorne, Kip
This slingshot maneuver is not seen or discussed in Interstellar, but the next one is mentioned, by Cooper: “Look, I can swing around that neutron star to decelerate,” he says. Deceleration is necessary because, having fallen under Gargantua’s huge gravitational pull, from the Endurance’s orbit to Miller’s orbit, the Ranger has acquired too much speed; it is moving c/4 faster than Miller’s planet. In Figure 7.3, the neutron star, traveling leftward relative to Miller’s planet, deflects and slows the Ranger’s motion so it can rendezvous gently with the planet.
Fig. 7.2. The Ranger performs a slingshot maneuver around a small black hole, deflecting it downward, toward Miller’s planet.
Fig. 7.3. Slingshot around a neutron star enables the lander to rendezvous with Miller’s planet.
Now, there is a feature of these slingshots that could be very unpleasant. Indeed, deadly: tidal forces (Chapter 4).
To change velocities by as much as c/3 or c/4, the Ranger must come close enough to the small black hole and neutron star to feel their intense gravity. At those close distances, if the deflector is a neutron star or is a black hole with radius less than 10,000 kilometers, the humans and the Ranger will be torn apart by tidal forces (Chapter 4). For the Ranger and humans to survive, the deflector must be a black hole at least 10,000 kilometers in size (about the size of the Earth).
Now, black holes that size do occur in Nature. They are called intermediate-mass black holes, or IMBHs, and despite their big size, they are tiny compared to Gargantua: ten thousand times smaller.
So Christopher Nolan should have used an Earth-sized IMBH to slow down the Ranger, not a neutron star. I discussed this with Chris early in his rewrites of Jonah’s screenplay. After our discussion, Chris chose the neutron star. Why? Because he didn’t want to confuse his mass audience by having more than one black hole in the movie. One black hole, one wormhole, and also a neutron star, along with Interstellar’s other rich science, all to be absorbed in a fast-paced two-hour film; that was all Chris thought he could get away with. Recognizing that strong gravitational slingshots are needed to navigate near Gargantua, Chris included one slingshot in Cooper’s dialog, at the price of using a scientifically implausible deflector: the neutron star instead of a black hole.
Intermediate-Mass Black Holes in Galactic Nuclei
A 10,000-kilometer IMBH weighs about 10,000 solar masses. That’s ten thousand times less than Gargantua, but a thousand times heavier than typical black holes. These are the deflectors Cooper needs.
Some IMBHs are thought to form in the cores of dense clusters of stars called globular clusters, and some of them are likely to find their way into the nuclei of galaxies, where gigantic black holes reside.
An example is Andromeda, the nearest large galaxy to our own (Figure 7.4), in whose nucleus lurks a Gargantua-sized black hole: 100 million solar masses. Huge numbers of stars are drawn into the vicinity of such gigantic black holes; as many as a thousand stars per cubic light-year. When an IMBH passes through such a dense region, it gravitationally deflects the stars, creating a wake with enhanced density behind itself (Figure 7.4). The wake pulls on the IMBH gravitationally, slowing the IMBH down, a process called “dynamical friction.” As the IMBH very gradually slows, it sinks deeper into the vicinity of the gigantic black hole. In this manner, Nature could provide Cooper, in my interpretation of Interstellar, with the IMBHs that he needs for his slingshots.19
Fig. 7.4. Left: The Andromeda galaxy, which harbors a Gargantua-sized black hole. Right: The dynamical friction by which an IMBH will gradually slow down and sink into the vicinity of the gigantic black hole.
Orbital Navigation by Ultra-Advanced Civilizations: A Digression
The orbits of planets and comets in our solar system are all ellipses to very high accuracy (Figure 7.5). Newton’s laws of gravity guarantee and enforce this.
By contrast, around a gigantic, spinning black hole such as Gargantua, where Einstein’s relativistic laws hold sway, the orbits are far more complex. Figure 7.6 is an example. For this orbit, each trip around Gargantua would require a few hours to a few days, so the entire pattern in Figure 7.6 would be swept out in about a year. After a few years, the orbit would pass near most any destination you might wish, though the speed at which you arrive might not be right. A slingshot might be needed to change speed and make a rendezvous.
Fig. 7.5. The orbits of planets, Pluto, and Halley’s comet in our solar system are all ellipses.
Fig. 7.6. A single orbit of a spacecraft or planet or star around a gigantic, fast-spinning black hole such as Gargantua. [From a simulation by Steve Drasco.]
I’ll let you imagine how an ultra-advanced civilization might use such complex orbits. In my science interpretations of the movie, for simplicity I mostly eschew them and focus primarily on circular, equatorial orbits (those of the parked Endurance, Miller’s planet, and the critical orbit), and on simple trajectories for the Endurance as it travels from one circular equatorial orbit to another. An exception is the orbit of Mann’s planet, discussed in Chapter 19.
NASA’s Gravitational Slingshots in the Solar System
Let’s return from the world of the possible (what the laws of physics allow) to hard-nosed, real-life gravitational slingshots in the comfy confines of our solar system (what humans have actually achieved as of 2014).
You may be familiar with NASA’s Cassini spacecraft (Figure 7.7). It was launched from Earth on October 15, 1997, with too little fuel to reach its destination, Saturn. The deficit was dealt with by slingshots: around Venus on April 26, 1998; a second slingshot around Venus on July 24, 1999; around Earth on August 18, 1999; and around Jupiter on December 30, 2000. Arriving at Saturn on July 1, 2004, Cassini slowed down with the aid of a slingshot around Saturn’s closest moon, Io.
Fig. 7.7. The trajectory of Cassini from Earth to Saturn.
None of these slingshots looked like the ones I described above. Instead of strongly deflecting the spacecraft’s direction of motion, Venus, Earth, Jupiter, and Io deflected it only mildly. Why?
The deflectors’ gravity was too weak to produce a strong deflection. For Venus, Earth, and Io, the deflection was inevitably small because their gravity is intrinsically weak. Jupiter has much stronger gravity, but a large deflection would have sent Cassini in the wrong direction; reaching Saturn required a small deflection.
Despite the small deflections, Cassini got substantial kicks from the flybys, big enough to compensate for inadequate fuel. In each flyby (except Io), Cassini traveled behind the deflecting planet but at an angle, so the planet’s gravity optimally pulled Cassini forward, speeding it up. In Interstellar, the Endurance does a similar slingshot around Mars.
Cassini has been exploring Saturn and Saturn’s moons for the past ten years, sending back amazing images and information—a treasure trove of beauty and science. For a glimpse, see http://www.nasa.gov/mission_pages/cassini/main/.
By contrast with these weak slingshots in the solar system, Gargantua’s intense gravity can grab even objects moving at ultrahigh speeds and throw them around on strongly bent slingshots. Even a light ray. This produces gravitational lensing, the key to seeing Gargantua.
* * *
19 The probability of finding IMBH’s at the needed locations and times is small, but in the spirit of science fiction, since it is within the bounds of physical law, we can utilize them.
8
Imaging Gargantua
Black holes emit no light, so the only way to see Gargantua is by its influence on light from other objects. In Interstellar the other objects are an accretion disk (Chapter 9) and the galaxy in which it lives including nebulae and a rich field of stars. For the sake of simplicity, let’s include only the stars for now.
Gargantua casts a black shadow on the field of stars and it also deflects the light rays from each star, distorting the
stellar pattern that the camera sees. This distortion is the gravitational lensing discussed in Chapter 3.
Figure 8.1 shows a rapidly spinning black hole (let’s call it Gargantua) in front of a field of stars, as it would appear to you if you were in Gargantua’s equatorial plane. Gargantua’s shadow is the totally black region. Immediately outside the shadow’s edge is a very thin ring of starlight called the “ring of fire” that I intensified by hand to make the edge of the shadow more distinct. Outside that ring we see a dense sprinkling of stars with a pattern of concentric shells, a pattern produced by the gravitational lensing.
Fig. 8.1. The gravitationally lensed pattern of stars around a rapidly spinning black hole such as Gargantua. When seen from far away, the shadow’s angular diameter, measured in radians, is 9 Gargantua radii divided by the observer’s distance from Gargantua. [From a simulation by the Double Negative visual-effects team.]
As the camera orbits around Gargantua, the field of stars appears to move. This motion combined with the lensing produces dramatically changing patterns of light. The stars stream at high speed in some regions, they float gently in others, and they’re frozen in still other regions; see the film clip on this book’s page at Interstellar.withgoogle.com.
In this chapter I explain all these features, beginning with the shadow and its ring of fire. Then I describe how the black-hole images in Interstellar were actually produced.
When imaging Gargantua in this chapter, I treat it as a fast-spinning black hole, as it must be to produce the extreme loss of time that the Endurance’s crew experience relative to Earth (Chapter 6). However, for fast spin, a mass audience could be confused by the flattening of the left edge of Gargantua’s shadow (Figure 8.1) and by some peculiar features of the star streaming and the accretion disk, so Christopher Nolan and Paul Franklin chose a smaller spin, 60 percent of the maximum, for their Gargantua images in the movie. See the last section in Chapter 9.
Warning: The explanations in the following three sections may require a lot of thought; you can skip them without losing pace with the rest of the book. Not to worry!
The Shadow and Its Ring of Fire
The shell of fire (Chapter 6) plays a key role in producing Gargantua’s shadow and the thin ring of fire alongside it. The shell of fire is the purple region surrounding Gargantua in Figure 8.2, and it contains nearly trapped photon orbits (light rays) such as the one in the upper right inset.20
Suppose you are at the location of the yellow dot. The white light rays A and B and others like them bring you the image of the ring of fire, and the black light rays A and B bring you the image of the shadow’s edge. For example, the white ray A originates at some star far from Gargantua, it travels inward and gets trapped on the inner edge of the shell of fire in Gargantua’s equatorial plane, where it flies round and round, driven by the whirl of space, and then escapes and comes to your eyes. The black ray also labeled A originates on Gargantua’s event horizon, it travels outward and gets trapped on that same inner edge of the shell of fire, where it goes round and round, then escapes and reaches your eyes alongside the white ray A. The white ray brings you an image of a bit of the thin ring; the black ray, an image of a bit of the shadow’s edge. The shell of fire is responsible for merging the rays side by side and directing them toward your eyes.
Fig. 8.2. Gargantua (central spheroid), its equatorial plane (blue), its shell of fire (purple and violet), and black and white light rays that bring you images of the shadow’s edge and the thin ring alongside it.
Similarly for the white and black rays B, except they get trapped on the outer edge of the shell of fire going clockwise (struggling against the whirl of space), while rays A are trapped on the inner edge going counterclockwise (and driven by the whirl of space). The flattening of the shadow’s left edge in Figure 8.1 and rounding of its right edge are due to rays A (left edge) coming from the inner edge of the shell of fire, very close to the horizon, and rays B (right edge) from the outer edge of the shell of fire, much further out.
Black rays C and D in Figure 8.2 begin on the horizon, travel outward and get trapped on nonequatorial orbits in the shell of fire, and then escape from their trapped orbits and come to your eyes, bringing images of bits of the shadow edge that lie outside the equatorial plane. The trapped orbit for ray D is shown in the upper right inset. White rays C and D (not shown), coming from distant stars, get trapped alongside black rays C and D, and then travel to your eyes alongside C and D, bringing images of bits of the ring of fire alongside bits of the shadow edge.
Lensing by a Nonspinning Black Hole
To understand the pattern of gravitationally lensed stars outside the shadow and their streaming as the camera moves, let’s begin with a nonspinning black hole and with light rays that emerge from a single star (Figure 8.3). Two light rays travel from the star to the camera. They each travel along the straightest line they can in the hole’s warped space, but because of the warping, each ray gets bent.
One bent ray travels to the camera around the hole’s left side; the other, around its right side. Each ray brings the camera its own image of the star. The two images, as seen by the camera, are shown in the inset of Figure 8.3. I put red circles around them to distinguish them from all the other stars the camera sees. Note that the right image is much closer to the hole’s shadow than the left image. This is because its bent ray passed closer to the hole’s event horizon.
Fig. 8.3. Top: The warped space around a nonspinning black hole as seen from the bulk, and two light rays that travel through the warped space from a star to the camera. Bottom: The gravitationally lensed pattern of stars that is seen by the camera. [From a simulation by Alain Riazuelo; for a film clip of his simulation, see www2.iap.fr/users/riazuelo/interstellar .]
Each of the other stars appears twice in the picture, on opposite sides of the hole’s shadow. Can you identify some of the pairs? The black hole’s shadow, in the picture, consists of directions from which no rays can come to the camera; see the triangular shaped region labeled “shadow” in the upper diagram. All the rays that “want to be” in the shadow got caught and swallowed by the black hole.
As the camera moves rightward in its orbit (Figure 8.3), the pattern of stars seen by the camera changes as shown in Figure 8.4.
This figure highlights two particular stars. One is circled in red (the same star circled in Figure 8.3). The other is inside a yellow diamond. We see two images of each star: one image is outside the pink circle; the other is inside the pink circle. This pink circle is called the “Einstein ring.”
As the camera moves rightward, the images move along the yellow and red curves.
The star images outside the Einstein ring (the primary images, let’s call them) move in the way one might expect: smoothy from left to right, but deflecting away from the black hole as they move. (Can you figure out why the deflection is away from the hole instead of toward it?)
Fig. 8.4. The changing star pattern seen by the camera as it moves rightward in its orbit in Figure 8.3. [From the simulation by Alain Riazuelo; see www2.iap.fr/users/riazuelo/interstellar .]
However the secondary images, inside the Einstein ring, move in an unexpected manner: They appear to emerge from the right edge of the shadow, move outward into the annulus between the shadow and the Einstein ring, swing leftward around the shadow, and descend back toward the shadow’s edge.
You can understand this by going back to the upper drawing in Figure 8.3. The right ray passes near the black hole, so the right stellar image is near its shadow. Earlier in time, when the camera was further leftward, the right ray had to pass even closer to the black hole in order to bend more strongly and reach the camera, so the right image was very close to the edge of the shadow. By contrast, earlier in time, the left ray passed rather far from the hole and so was nearly straight and produced an image rather far from the hole.
Now, if you’re ready, think through the subsequent motions of the images, depicted in Figure 8.4.
Lensing by a Rapidly Spinning Black Hole: Gargantua
The whirl of space generated by Gargantua’s very fast spin changes the gravitational lensing. The star patterns in Figure 8.1 (Gargantua) look somewhat different from those in Figure 8.4 (a nonspinning black hole), and the streaming patterns differ even more.
For Gargantua the streaming (Figure 8.5) reveals two Einstein rings, shown as pink curves. Outside the outer ring, the stars stream rightward (for example, along the two red curves), as they did for a nonspinning black hole in Figure 8.4. However, the whirl of space has concentrated the stream into narrowed high-speed strips along the back edge of the hole’s shadow, strips that bend somewhat sharply at the equator. The whirl has also produced eddies in the streaming (the closed red curves).
The secondary image of each star appears between the two Einstein rings. Each secondary image circulates along a closed curve (for example, the two yellow curves), and it circulates in the opposite direction to the red streaming motions outside the outer ring.
Fig. 8.5. The star streaming patterns as seen by a camera near a rapidly spinning black hole such as Gargantua. In this simulation by the Double Negative visual-effects team, the hole spins at 99.9 percent of the fastest possible, and the camera is in a circular, equatorial orbit with circumference six times larger than the horizon’s circumference. For a film clip of this simulation, see this book’s page at Interstellar.withgoogle.com.
