This chapter describes multidimensional hydrodynamics, and it contains four sections. The first section discusses classical multidimensional shock-capturing hydrodynamics and the application of modern high-order Godunov-type methods that reduce the problem into a class of almost engineering tasks.

Scientific input includes equations of state, reaction rates, and kinetic coefficients, described in the second section. The main difficulties of the kinetic Boltzmann approach discussed in the previous chapter are not only the multidimensionality of the phase space but also the calculation of the reaction rates. These reaction rates usually require using implicit schemes in the case of nontransparent regions, and multidimensional problems become very hard to study. The key point of this section is the proposal to move from the kinetic Boltzmann treatment in 7D phase space (r, p, t) to a hydrodynamic one with diffusion and flux limiters in 5D phase space (r, ϵ, t). The diffusion with flux limiter approach uses some free parameters for the interpolation of spectral energy fluxes in the intermediate case between the transparent (free flow) and the nontransparent (diffusion or heat conduction) regions. The first calculations within such an approach were performed in [131] for the gravitational collapse with the neutrino transport in spherically symmetric case. In such a problem one has to carry out advection by explicit code. At the same time, diffusion of spectral energies is carried out by an implicit scheme, such as the Crank-Nicolson method [100]. Finally, reaction rates are computed using an implicit scheme for the system of ODEs. In the multidimensional 2D or 3D cases the splitting on the time integration along separate directions is made. Given the limited space resolution in the 3D case in comparison with the 1D one, high-order Godunov-type methods are required.

The third section discusses the Riemann problem solver within special relativity. The numerical relativity does not offer universal receipts. Moreover, the current state of numerical relativity in strong gravitational fields is more similar to art than to science. Therefore, in this section, only special relativistic high-order shock-capturing methods are discussed.

The fourth section briefly describes particle-based simulations, in particular, the smoothed-particle hydrodynamics method. Such mesh-free methods are an attractive alternative to traditional grid-based hydrodynamics.

The endeavor of writing this book started from a series of lectures given by the first author for students of the International Relativistic Astrophysics PhD program (IRAP PhD) supported by the Erasmus Mundus program of the European Commission. For this book the material has been expanded and more topics incorporated. It soon became clear that an updated and systematic presentation of relativistic kinetic theory and its numerous applications in astrophysics and cosmology is lacking in the literature. Some existing monographs, presenting fundamental aspects of kinetic theory, are focused on selected applications. Others, which contain applications of kinetic theory in relativistic astrophysics and cosmology, lack the presentation of fundamental concepts of relativistic kinetic theory. Moreover, none of the existing monographs discussed in depth various numerical methods developed and successfully applied in kinetic theory in the recent decades. This last observation urged us to bridge this gap in the literature. This effort eventually resulted in the current monograph, divided in three parts. Parts I and III, with the sole exception of the last chapter, were written by the first author. Part II and the last chapter of Part III were written by the second author.

My first glimpse of Titan in real time wasn't through a toy telescope or even through the respectable telescopes that one finds at star parties. It was a hot August in 1995 and I was at Palomar Observatory during Saturn ring plane crossing, that rare time when the bright rings appear edge-on and nearly disappear to reveal the whole of Saturn's six major inner moons, encircling the planet like a string of pearls, suspended on some imaginary scaffolding in the skies. The biggest gem of all was Titan, a ghostly white orb silently looming in space, lit with the soft reflected light of the Sun, inviting us to come closer (Figure 7.1). I had seen much better images from Voyager 1 depicting this mysterious, cloud-enshrouded world, but never had the moon been so palpably close as during this live-action view. I could not have imagined the enchanted world that lurked beneath those clouds.

Titan was discovered in 1655 by Christiaan Huygens (1629–1695), a Dutch astronomer and mathematician who designed and constructed technologically advanced telescopes, which again shows how technology drives science. He is perhaps best known for his discovery that the velocity of light is finite. Huygens even built a refracting telescope with its lenses in the open air – no unwieldy tubes to drag it down (Figure 7.2). But Huygens's clever invention wasn't very practical: it was just too difficult to align, and there was no way of keeping out stray ambient light. Huygens's great discoveries were made with more pedestrian “bread and butter” telescopes. Titan, for example, was discovered with a 12-foot, 50 power telescope that is today surpassed by the typical equipment in a college's observatory.

Most moons of the Solar System appear as fading tenuous blinks of light in the sky, even when seen through a moderately sized telescope. Intently studying these tiny points, astronomers have used their clever tricks through the ages to infer a surprising amount of information. For example, if a rotating moon or planet dims and brightens in a regular fashion, one can be sure – at least if the body is round – that one hemisphere contains materials that are much brighter than the other – perhaps patches of frost.

I clearly remember a conversation I had with my brother Bruce when I was six or seven in the small bedroom we shared in our home in the steel town of Bethlehem, Pennsylvania. My brother asked me how long I thought it would take to freeze to death if you were standing on Pluto and weren't wearing a spacesuit. Pluto was one of the nine major planets back then, and as the farthest out, it was the coldest thing a child could imagine. I asked Bruce if he thought Pluto was colder than Antarctica: we both thought it was. We figured if you were bundled up in a snowsuit and covered up with blankets, you might live five minutes at the South Pole. This era was pre-internet – even hand-held calculators were more than a decade away – so we couldn't look up the temperature of Pluto, or even of Antarctica. We finally decided Pluto was so cold you'd die in about five seconds. I remember the view I had of the surface of little Pluto – ice everywhere and very dark. My trusty Child's Book of the Stars had only six sentences on Pluto, and its stated size of 4,000 miles was disappointingly small, only about half the size of the Earth. But as we shall see, Pluto shrank even further before its planetary demotion.

Pluto grabbed me again in 1965, at the same time Mariner 4 sent back its images of a disappointing, cratered, moonlike Mars. Scholastic Book Services, the lifeline of curious children, offered a 45-cent book called The Search for Planet X by Tony Simon. Sputnik had been launched eight years earlier, the space race was in full swing, and everyone was looking up. The Search was pure romance, offering the improbable story of a farm boy who discovered a planet.

Clyde Tombaugh was drawn to the stars by the wide and clear skies of Kansas. He built his own telescopes, from grinding the lenses to fashioning their mounts out of old parts from farm machinery, including a cream separator.

To the casual skywatcher Mercury appears near the horizon just after sunset as a faint orange star bathed in the fading glow of the western sky. To the more dedicated observer, the planet also appears right before sunrise in the eastern sky. The ancients had two names for its dawn and dusk appearances: Apollo in the morning, to signify the appearance of the Sun, and Hermes in the evening, to acknowledge the Greek messenger god. The speed of Mercury's motion in its orbit – and as seen from the Earth – is faster than the other five planets easily visible to the naked eye. By the fourth century BCE, during the golden age of Greek experimental science, astronomers noticed that this faint planet appeared in the same position relative to the Sun at both dawn and dusk, and they realized the two apparitions were the same body. The Romans named the planet Mercury after their own swift messenger god. In Nordic mythology, Mercury was associated with Odin, or Wodin, from which Wednesday (Mercredi in French with similar renditions in the Romance languages) is derived.

Many astronomers have never seen Mercury, and the first sighting of this elusive, “mercurial” planet is always memorable. I still remember the night over a half-century ago when I stood alone in the middle of a corn field near my parents’ house in Bethlehem, Pennsylvania and compared the great night sky to a tiny map I had cut out of the Bethlehem Globe Times. The Sun had shed its last ray, and I felt so small as I stood where the soft cusp of the field gave way to the harsh vastness of space. But I was reassured when I saw the little planet, blinking on and off, unmistakably where it should be.

Little experimental triumphs such as this one, when the smallness of our world and our concerns are dwarfed by the immensity and predictability of the stars and planets, were what drew me to the study of the cosmos. I didn't see the planet again with my own eyes until the mid-1990, when I was a fully-fledged astronomer observing on the 200-inch Hale telescope at Palomar Mountain. My colleague Phil Nicholson of Cornell University and I went out onto the catwalk circling the dome to inspect the weather and observing conditions.

On the drive up Interstate 15 from Los Angeles towards Las Vegas, as the land begins to ascend and give entry to the city of infinite dreams and dross, one can take a detour from the journey and turn at the small town of Baker (“Gateway to Death Valley”). The environment that continues north along the scenic California State Route 127 is almost indistinguishable from that of Mars, except for the few straggly plants that hug the road and other places with just a little moisture. Vast ranges of sanded plains greet jagged mountains and dry lake beds. The terrain is so Mars-like, that a mock-up of the Mars Science Laboratory Curiosity was tested there in 2012, a year after the actual rover was launched (Figure 3.1). Engineers moved the car-sized vehicle over the desert to see how its wheels would fare in the sandy soil of Mars. Spirit, one of JPL's earlier Mars Exploration Rovers, had gone to its final resting place in 2009 after one of its wheels got stuck in the martian sand after more than six years of studying the surface of Mars. The other Rover, Opportunity, is still going strong at the time of writing.

The cautious and ingenious engineers at JPL weren't going to let a similar fate overtake Curiosity. Their mock-up, with a mass of only about 38% of the real thing to account for the lower gravity on Mars, was beset with challenging situations similar to those expected on Mars. It was driven up and over the sides of sand dunes, and into the deepest, softest sand imaginable. The wheels of Curiosity were engraved with giant treads with immense gripping power, and in a flight of flair and fancy, the prototype was engraved with the letters JPL. When NASA Headquarters saw that they banned it from the final model. Under the “one NASA” policy, the entire Agency had to share in the glory. But the clever engineers at JPL struck back: they embedded the Morse code symbols for JPL in the tire, so the true geeks’ code is engraved in endless trails all over the martian sands.

The four main moons surrounding Jupiter have played an outsized role in the history of astronomy, just as Mercury has. They comprise a sort of mini solar system, orbiting in regular orbits about the massive Jupiter. If they were separated from the bright glare of Jupiter, they could all be seen with the naked eye, and there are occasional stories of sharp-eyed individuals being able to see the moons. They range in size from a little smaller than the Moon to larger than Mercury. The four moons seem to have no kinship with each other, spanning the range of celestial personalities from the turbulent Io to the calm Callisto. As is the case with our own Moon, they are tidally evolved and keep the same face toward Jupiter, in a state astronomers call synchronous rotation. In honor of their discoverer, Galileo Galilei, they are called the Galilean moons.

Just as I shall always remember my first view of Mercury, I shall never forget the first time I saw some of Galileo's moons. I was in the fifth grade, and my parents had just bought me a simple little telescope from Hess's, the local department store. It was a reflecting telescope with a 4-inch mirror, magnifying celestial objects only a little bit better than a pair of binoculars. The tube was made of flimsy black cardboard. I eagerly put the telescope together and placed it out on the front lawn. After I secured its wobbly tripod, I pointed it to Jupiter, which was rapidly declining in the west. The Moon wasn't up – otherwise I would have looked at it first – so the sky was dark and Jupiter loomed brightly, seemingly propped up by the trees lining the horizon. Jupiter is the third most luminous object in the night sky, after the Moon and Venus. After some frustrating fiddling, the bright planet lurched into the field of view of my little instrument. I wasn't able to see the Great Red Spot, Jupiter's eternal hurricane, but right next to Jupiter were two tiny dots, pulsating in Earth's turbulent atmosphere and in the scattered light from Jupiter.

For a few short moments in my life I thought I had something to teach the Teacher of the Dalai Lama. I was in my fourth year of graduate school at Cornell, settling in on my PhD thesis topic to understand what the surface of Europa is like. I was busy analyzing some Voyager 2 images of the surface of Europa in the Spacecraft Planetary Imaging Facility (SPIF). The Teacher of the Dalai Lama was coming to Cornell to see Carl Sagan, and he wanted a little tour of SPIF. Since I knew a lot about SPIF – I practically lived there at that time – Sagan had tasked me with giving the tour. Enrobed in cascades of maroon and gold, the ethereal monk glided into SPIF with an entourage of intimidating British interpreters, scholars, and disciples. I thought the Teacher would share my excitement about this ice world in space. I told him how smooth the surface was, how there was probably an ocean beneath its cracked icy surface, and – jokingly – that it might be fun to ski on Europa. I had heard that the Dalai Lama had a sense of humor, but his teacher apparently not so much. He listened politely to my lecture, but he looked at me sternly after I had finished, and he said (at least his interpreter said): “There are many worlds in this Universe that you cannot even envision.”

The Teacher was of course correct, even in the scientific sense. At that time, not a single planet outside our Solar System had been discovered, but we thought there must be billions. Exoplanets, or extrasolar planets – planets around other stars, and extrasolar systems – planetary systems around other stars – were hypothesized to not only exist but to abound in the Universe. A famous set of images taken by the Hubble Space Telescope, the deep field surveys, drives home the immensity of our Universe and the possible number of worlds within it. Figure 10.1 shows the Hubble Ultra Deep Field survey, constructed of hundreds of images with a total exposure time of nearly one million seconds and covering one 13-millionths of the sky.

There is an apocryphal, but famous, story that has circulated for years among planetary scientists and others. When confronted by two Yale scholars with evidence of stones falling to the Earth after a giant fireball appeared in the sky near Weston, Connecticut in 1807, Thomas Jefferson supposedly said that “It is easier to believe that two Yankee professors would lie than that stones would fall from heaven.” The actual comments made by Jefferson were more temperate, reflecting the normal skepticism of any scientist confronted with unusual new data. In a letter dated February 15, 1808 to Daniel Salmon, Jefferson writes:

Other historical falls of meteorites include a devastating event in CE 1490, when Chinese records state that stones from the sky weighing several pounds fell “like rain” and killed 10,000 people in what is now Ganzu province. A large tsunami accompanied by fires, which can be caused by an asteroid impacting the oceans, was recorded in both New Zealand and Australia around 1500. (The appearance of Comet C/1490 Y1 and possibly associated yearly meteor showers is probably unrelated as it was observed about four months before the meteor fall.) No surviving meteorites from this event have been found in China. Could it have been a deadly hailstorm with exaggerated casualties? We just don't know.

Even more uncertain is the biblical story of stones that fell on the Amorites: “It happened when they fled before Israel. On the descent of Bet-Horin, God threw upon them large stones (avanim g'dolot) from heaven towards Azekah, and they died. The number that died from the stones of hail (avnai-barad) was greater than the number slain by Israel by the sword” (Joshua 10: 11). The first phrase seems to suggest a meteor shower, but the second phrase explicitly mentions the more common hail. In his book Rain of Iron and Ice, planetary scientist John S. Lewis suggests that the halting of the Sun two verses later (Joshua 10: 13) could be an illusion born of a brightly lit sky, similar to that observed after modern, confirmed meteor strikes.

Writings of the great thinkers abound with words expressing the great hold of astronomy. Plato said “astronomy compels the soul to look upwards and leads us from this world to another.” When William Herschel (1738–1822) – the father of modern observational astronomy – received the Royal Society's Copley Medal in 1781, the Society President and naturalist Joseph Banks, stated that “the treasures of heaven are well-known to be inexhaustible.” Astronomers themselves have spoken of their unquenchable curiosity and their drive and persistence to slake that curiosity. German astronomer Johann Schroeter (1745–1816) spoke of the “impulse to observe,” while another astronomer said the purpose of existence is to observe. When Herschel was once asked why he had become an astronomer (with the implication – familiar even today – that a life of observing is impractical, even useless) he simply said that when he looked up and saw the beauty and wonder of the skies he didn't understand why everyone wasn't an astronomer.

But then there is the counterpoint in the public mind, captured in Walt Whitman's (1819–1892) poem “When I Heard the Learned Astronomer”:

The calculating astronomer misses the essence of the thing. As brilliant as he was, Whitman was wrong on this one. Most non-scientists think science is dry, fact-based, memorization – exact, or impenetrable. It is none of those things. Science is an endeavor of creative thought and activity, and it affects our everyday world by paving the way for technological inventions and by providing the groundwork for everything from weather forecasting to curing cancer. In its highest form it is no different from poetry.

Whenever I give a public talk on space exploration, or speak with someone casually, such as on an airplane or on a bus, I am often asked why NASA isn't doing more: “Why don't we have bases on the Moon? Why aren't there astronauts on Mars, and why didn't New Horizons go into orbit around Pluto or send back a sample?” My answer is invariably, “we would like to do all those things, if only NASA had the funds. We can think of all sorts of marvelous missions, but our budget is so constrained. We are limited to just a few high-priority projects.”

In the past we were able to do so much more, but as NASA's funding dwindled from 4.5% of the national budget during the peak years of the Apollo lunar program, to less than 0.5% today, we are confined to a bare-bones portfolio of exploration. The dreams of my childhood, and of so many others, have been diminished as the budget for scientific work in general has declined. It is not only NASA. IBM also used to have an outstanding group of researchers, and of course there was Bell Laboratories, which spawned so many Nobel Prizes, including the work of Arno Penzias and Robert Wilson with their discovery in 1964 of the remnant thermal radiation from the Big Bang, the Universe's moment of creation almost 14 billion years ago. Less commonly someone will say “Why spend all that money out in space?” I will respond: “None of the money is spent in space. It is all spent here on Earth providing good jobs that cannot be outsourced or eliminated.”

New Horizons will continue on to the smallish Kuiper Belt Object 2014 MU69, Juno started exploring the interior of Jupiter in mid-2016, the Mars Atmosphere and Volatile EvolutioN Mission (MAVEN) is orbiting Mars to understand how it lost its atmosphere, and the InSight Lander will explore the interior of Mars with seismometers and heat-flow detectors. Mars 2020 will follow on the heels of the Mars Curiosity Lander to explore possible habitable environments on Mars and to search for life. But it will only cache a collection of samples for later return because we don't have the funds – or the technological resources – to return this valuable cargo in the near future.