The Lake County Astronomical Society

The formula for net force of emitted photons acting on the asteroid (day side force minus night side force is: [(8/3)(σπr²T⁴)/(c)][ΔT/T].[14] With this formula, we can use some numbers from a real asteroid of 12 km in diameter (big enough to cause a mass extinction) to yield an approximate net acceleration of 6.5 x 10^-15 (m/sec²).[15] At any given time the acceleration is very slight, but over millions of years, the constant acceleration provided by the Yarkovsky effect builds the velocity and the asteroid is moved an appreciable distance.

The computer model of Farinella and Vokrouhlicky (1999) showed that an asteroid 20 km in diameter would see a semi-major axis shift of .01 AU per billion years.[16] Even such a small change is significant because it can be sufficient to push the asteroid into a Kirkwood gap. In the late 19^th century an astronomer, Daniel Kirkwood, noted that there are certain orbits in the asteroid belt that are curiously devoid of material. These turned out to be a sub-group of the orbits that are in integer ratios with the orbit of Jupiter, such as 3:1, 5:2, 7:3, 2:1. These ratios mean, for example, that the object orbits the sun 3 times for every 1 time that Jupiter does so. An asteroid in these particular regions will be subject to a resonance: a periodic disturbance from the gravity of Jupiter that causes changes in the orbit of the asteroid. The resonance can force the asteroid towards the inner solar system. Further interactions with the gravity of Mars may eventually lead to a collision with Earth. So the Yarkovsky effect is relevant to mass extinctions on Earth because it can nudge asteroids into a Kirkwood gap from which the asteroid is ejected by Jupiter's gravity, leading to a potential collision with Earth.[17]

J.N. Spitale has proposed that the Yarkovsky effect could be exploited to avert the collision of an asteroid with the Earth.[18] Painting an asteroid black or otherwise changing its reflection and/or absorbtion properties vis-a-vis photons could slowly move it off a collision course. The technique would work only with relatively small asteroids, but could still save a city from becoming a crater. The reason it would work on only smaller asteroids is that as the radius the object increases, its volume (4/3πr³), and therefore its mass, increases much more rapidly than its surface area (4πr²). For a large asteroid, there is very much more mass to move, but not so much more surface area to radiate the photons.

Other explanations of the causes of Earth impacts focus on comets. Comets may be the cause of the majority of the Earth's impact craters that are over 50 km in diameter.[19] These large craters are most likely to have caused a mass extinction.

A comet is said to be "perturbed" when it is disturbed from its dormant state and nudged towards the inner solar system or out into interstellar space. When the comet enters the inner solar system and is heated by the sun sufficiently, it begins to have a tail (actually up to 3 separate tails: gas, ion and dust)[20] and is then said to be "observable." If it is determined that it is the first time the comet has journeyed to the inner solar system, it is said to be a "new" comet.

The question is, what causes comet perturbations? Two hypotheses, 'Planet X' and 'Nemesis' were prominent the 1980s. 'Planet X' proposes that a yet unobserved Planet X disturbs the Kuiper belt of comets,[21] dislodging some into an intercept course with Earth. It is these comets which cause mass extinctions when they collide with the planet. Nemesis proposes that an unobserved companion to the sun, probably a dwarf star with a highly elliptical orbit, periodically passes near enough to the Oort cloud of comets[22] to yank some from their orbits and send them towards Earth.[23] Eminent Paleontologist David M. Raup has charged that these hypotheses are ad hoc.[24]

ad hoc hypotheses; these are hypotheses designed to explain a legitimate set of geological data and associated findings. That dataset was Earth's known impact craters for approximately the past 600 million years. The findings were statistical analyses reporting that there was a 28 or 31 million year regularity (periodicity) in the cratering record.[25], [26]

One problem with 'Planet X' and 'Nemesis' is that they were explaining an incomplete geological data set. Since 1984, we have discovered many more impact craters (see footnote 1-3). Second, the dating of the craters was and is (to a lesser extent with improved technology and more research) uncertain. Third, even after 20 years of additional statistical analysis, there is still vigorous debate over whether there is a periodicity to the terrestrial cratering record.[27] However, whether or not the impacts occur on any regular cycle, as long as it is established by specific investigation of each crater and the global biological and geological evidence in that layer of sediment that at least 4 impact events are linked to mass extinctions, then it is worth enquiring into the causes of these impacts. If there is no periodicity to the impacts, it may not be useful to look for one sole cause of bolide impacts on Earth. That is the reason this review article looks at several possible causes of the impacts that contribute to mass extinctions.

There is a recent update to the solar companion genre of hypotheses, this time with a firm basis in astronomical observation. In the late 1990s, several researchers proposed the existence of a solar companion in an orbit within the Oort cloud. One group (Matese, Whitman and Whitmire) looked at about 30 comets with anomalous orbital characteristics. They projected the path of the comets back to aphelion (the point at which a comet is farthest from the sun). They proposed that the pattern of the comets' aphelia traces the near-circular orbital path of a large planet. The gravitational field of the planet perturbs comets along its orbital path. This planet would have a mass about 3 times that of Jupiter. It would be orbiting in the middle of the Oort cloud, about .4 light years (25,000 AU) away from the sun. It should be detectable by the Space Infrared Telescope Facility, to be launched in 2003.[28] The Matese group does not mention whether there may be a 28 or 31 million year periodicity to the proposed planet's orbit. Horner and Evans (2001) critically examine the Matese group's proposal and write, "We conclude that Matese et al.'s planet is a possible, perhaps even likely, explanation of the unusual pattern in the data ... Matese et al.'s planet warrants further and serious consideration."[29]

A thorough article has considered whether passing stars perturb the Oort cloud. Sanchez et al. (1999) look at the paths of nearby stars and project that in 1.5 million years the star Gliese 710 (a dwarf star of .6 solar masses) will pass within 1 light year of Earth and actually enter the Oort cloud. Its gravitational influence will eject about 2 million comets into an earth crossing orbit. The arrival of these comets will be spread over about 2 million years, meaning there will be only one extra comet per year. So, the Earth would probably not experience a perceptible increase in comet activity. The authors do not claim any connection to a mass extinction on Earth. Yet one of these 2 million comets might actually impact the Earth rather than merely pass close by. The probable reason the authors do not address this the impossibility of predicting specifically which comets will be ejected from the Oort cloud and sent towards Earth, and which of those will actually collide with the planet. The article looked 10 million years backward and 10 million years forward, and Gliese 710 is the best candidate they found for an Oort cloud perturber.[30]

We are focusing on "live" comets that still display a tail when they pass near the sun. A significant portion of the Earth-crossing asteroids may actually be asteroid-like "extinct" comets. These are rocky/metallic objects whose orbits have comet-like characteristics. They are thought to be comets that have lost all their gas, dust and water during repeated close passages around the sun. We are not considering extinct comets here, because generally these are only 1 km in diameter- not large enough to cause a mass extinction.[31]

4. Comets: The Galactic Tide

Another cause of comet impacts is the galactic tide. Galactic tide research describes how each comet in the Oort cloud is subject to a differential (net) gravitational force at each point in its orbit. To see how this is a tide, consider that the ocean tides on the Earth are caused by net gravitational forces in the sun-Earth-moon system. Meanwhile, the galactic tide is caused by the net gravitational forces in a sun-comet-galaxy system. The galactic tide is shifts in the orbits of comets, the earth tide is the rise and fall in sea level; both are caused by net gravitational force.

There are two components of the galactic tide: the radial tide and the disk tide (or z-tide). Together these are responsible for as much as 90% of the observable Oort cloud comets.[32] The radial tide accounts for 33% of these[33] while the disk tide accounts for 67%. While the disk tide[34] is dominant, here I explain the radial tide because it occurs in a simple flat x-y plane. Since 30% of the large (over 20 km in diameter) impact craters on Earth are ascribed to Oort cloud comets,[35] the radial tide is responsible for 10% of these craters. These percentages are open to debate. For example, a passing star may increase the number of observable Oort cloud comets by 50%.[36] So at that time, we could not attribute 90% of the Oort cloud comets to the tide. However, the galactic tide is most probably the dominant continuous factor in the baseline rate of observable Oort cloud comets.[37]

The radial tide works as follows: the gravitational force of the entire mass of the galaxy interior to our solar system acts as a point-source which gently alters the orbits of dormant comets, setting in motion events that can lead to a comet collision with the Earth. Understanding the galactic radial tide requires an overview certain aspects of solar system dynamics. The sun is about 2/3 of the way to the outer edge of the Milky Way galaxy, as shown in figure 3. The location is about 27,000 light years from the galactic center. The solar system oscillates vertically with amplitude of about 114 light years and a period of 60-90 million years.

The arrows indicating the vertical oscillation are exaggerated due to the difficulty of rendering such a small fraction of the horizontal radius (114 compared to 27,700).

After figure 12-6, Michael A. Seeds,Horizons, 6th edition, Pacific Grove: Books/Cole, 2000). Illustration credit: Nguyet Mai Vuong.

Figure 3 allows an explanation of the galactic disk tide. The galaxy is structured like a disk with a spherical center (consider figures 3 and 6 together). The "galactic plane" is a flat line running horizontally through the center of the disk in figure 3. When an object is inside the comparatively flat disk, it is said to be in the galactic plane. The disk tide refers to the gravitational force of all the matter in the galactic disk influencing the orbit of a comet. As the solar system oscillates above and below the galactic plane, the solar system's comets move with it, and the disk tide is operative. The gravity of the sun is so weak at Oort cloud distances that the gravity of the galactic disk can change the orbit of comets.

The mass of the entire Milky Way is 10¹¹ solar masses, most of which is located interior to our solar system because the inner part of the galaxy has a much denser concentration of matter. Our solar system is surrounded by the Oort cloud [see footnote 22]. Because the sun's gravity is so weak at Oort cloud distances, the gravity of other objects in the galaxy can have a significant effect on Oort cloud comets. In this instance we are considering the effect of the gravitational force of all the mass of the galaxy interior to our solar system on Oort cloud comets. The galaxy is essentially a collection of particle-like objects: stars, dust clouds and planets. This multitude of particles can be approximated as a sphere with all its gravitational force concentrated in a point at the center.[38] Thus we model the gravitational force of the mass of the galaxy interior to our solar system as emanating from the galactic center. This reduces the analysis to just three bodies: the sun, an Oort cloud comet, and the galactic center of gravity. The comet is orbiting the sun, and we are analyzing the effect of the sun's and the galactic center's gravity on this orbit.[39]

The sun exerts a gravitational force on the comet along the radius between the sun and comet. Taken alone, this force exerts no torque because the angle between r and F is 0.[40] Simultaneously, the galactic center of gravity exerts a gravitational force along the radius between it and the comet. Again, the angle between r and F is 0. However, the sum of these two forces produces a net force vector in the angle between the two forces (parallelogram rule of vector addition). This net force is at an angle with the sun-comet axis of rotation "r," so Φ (phi) does not equal zero and we have a torque. It is a "torque" and not a "force" because torque is the rotational analogue of force: the net gravitational force on the comet is acting at an angle. The angle and magnitude of the net force varies continuously as the comet moves through its orbit, so the magnitude of the torque varies as well. At any given point, the situation can be represented in the manner of a torque diagram found in a typical university physics textbook:

An Oort cloud comet is considered part of the solar system; it is (weakly) gravitationally bound to the sun.

Therefore, the sun is the axis of rotation and position vector "r" is rooted at the sun.

Φ is the angle between r and F when the two vectors are placed tail to tail (slide the F vector so that its origin is the sun).

The torque (symbolized by T) changes the angular momentum of the comet (L), since T=dL/dt.[41] The change in angular momentum changes the orbit of the comet according to the formula L=[GMa(1-e²)]^1/2: where G=the universal gravitational constant; M=the mass of the sun; a=the length of the semi-major axis and e=the eccentricity of the orbit. Since M and G are constants, only a and e can change. Hence, the gravitational force changes changes the orbital elements a and e. The perihelion distance q is given by q=a(1-e). Simply following the equations demonstrates that the galactic radial tide can change the perihelion distance of an Oort cloud comet.

The torque we are describing is not an overwhelming force which yanks a comet out of its orbit. It is a subtle force which works on an Oort cloud comet over thousands of orbits. Since each orbit of an Oort cloud comet may take 2-3 million years, the galactic tidal process works over a period of billions of years.

Depending on the original position of the comet and the cumulative effect of the torque, a particular comet's perihelion can move closer to the sun or farther away from the sun. This is why the phenomenon is called the radial tide. Radial motion is the motion of an object directed straight towards or away from a given vantage point, here the sun. When an Oort cloud comet is moved closer to the sun, it may enter the inner solar system, and then may collide with the Earth. Specifically, the comet may be caught by the gravity field of one of the gas giants and then may be forced towards the inner solar system, where it becomes observable. As noted above, about 10% of the large impact craters on Earth may be due to the galactic radial tide.

Conclusion

This article has sought to explain several areas of comparatively recent astrophysical research linking extraterrestrial phenomena to terrestrial mass extinctions. These include the Yarkovsky effect, the solar companion hypothesis, passing stars, and the galactic tide. These very long-term and exquisite astrophysical processes range in scale from the quantum to the immensely large. They appear to have caused asteroid and comet impacts on Earth and through these, they have led to the very existence of the human species.

[1] The extinction of the dinosaurs is known as the Cretaceous-Tertiary boundary mass extinction and is well-covered in textbooks, encyclopedias and various sites on the world wide web. Specific findings on other mass extinctions of life are: (1) Both correlation and causation between an asteroid or comet impact and the Eifelian-Givetian mass extinction (~380 million years ago (Ma)). [Ellwood, Benoist, El Hassani, Wheeler and Crick, "Impact Ejecta Layer from the Mid-Devonian: Possible Connection to Global Mass Extinctions," Science300 (2003): 1734- 1737.] Evidence also points to an impact role in: (2) the Permian-Triassic boundary (251 Ma). This was the most extensive mass extinction known, killing about 90% of all marine and 70% of all land life.¹ (3) the Triassic-Jurassic boundary (208 Ma), which is associated with the rise of the dinosaurs.² There are over 160 known surviving impact craters around the globe, several with diameters over 80 km.³ Ongoing projects are using remote sensing to detect additional craters despite the erasing processes of erosion, sedimentation, volcanism and plate tectonics.⁴

¹ Luann Becker et al., "Impact Event at the Permian-Triassic Boundary: Evidence from Extraterrestrial Noble Gases in Fullerenes", Science291 (2001): 1530-1533; G.J. Retallack, S. Abbas, E.S. Krull, "Search for evidence of impact at the Permian-Triassic boundary in Antarctica and Australia," Geology 26 (1998): 979-982; J.L. Isbell, R.A. Askin, G.J. Retallack, "Search for evidence of impact at the Permian-Triassic boundary in Antarctica and Australia: comment and reply," Geology 27 (1999): 859-860; K. Kaiho, et al., "Search for evidence of impact at the Permian-Triassic boundary in Antarctica and Australia," Geology 29 (2001).

² P.E. Olsen, et al., "Ascent of Dinosaurs linked to an Iridium Anomaly at the Triassic-Jurassic Boundary," Science 296 (2002): 1305-1307; D.M. Bice, et al., "Shocked Quartz at the Triassic-Jurassic Boundary in Italy," Science 255 (1992): 443; A.J. Mory, et al., "Woodleigh, Carnarvon Basin, Western Australia: A new 120 km diameter impact structure," Earth and Planetary Science Letters 117, no. 1-2 (2000): 119-128. But see, R.A. Kerr, "Did Volcanoes Drive Ancient Extinctions?" Science289 (2000): 1130-1131.

³ See The Earth Impact Database: http://www.unb.ca/passc/ImpactDatabase/CIDiameterSort.html

⁴ H. Koshiishi, et al., "New Information on Craters from Remote Sensing Data," in Csaba H. Detre, Terrestrial and Cosmic Spherules, (Budapest: Akademiai Kiado, 2000). This group has found a corresponding crater for the Triassic-Jurassic boundary.

For a review of mass extinctions and impact craters see, C. Koeberl, "The Sedimentary Record of Imact Events" at 360-364, chapter 18 in: Peucker-Ehrenbrink and Schmitz, eds. Accretion of Extraterrestrial Matter Throughout Earth's History, (New York: Kluwer/Plenum, 2001). Earlier sources include: M.R. Rampino, B.M. Haggerty, "Extraterrestrial impacts and mass extinctions of life," in: T. Gehrels ed., Hazards due to comets and asteroids, (Tucson: University of Arizona, 1994); M.R. Rampino, B.M. Haggerty,. "Impact crises and mass extinctions: a working hypothesis," in: G. Ryder, D. Fastovsky, S. Gartner eds., Special Paper 307: The cretaceous-tertiary event and other catastrophes in Earth history, (Boulder, CO: Geological Society of America, 1996); D. Raup, J. Sepkoski, Periodicity of extinctions in the geologic past. Proceedings of the National Academy of Science U.S.A. (1984): 81, 801-805. For a review of how comet impacts might cause mass extinctions of life see, E.M. Shoemaker, "Impact Cratering Through Geologic Time," Journal of the Royal Astronomical Society of Canada, vol. 92 no. 6 (1998).

[2] David M. Raup, The Nemesis Affair, (New York: W.W. Norton, 1986).

[3] This observation was made by eminent geologist and paleontologist David M. Raup in his book The Nemesis Affair (1986, at 20) and bears explanation. The current standard theory is that mass extinctions wipe out the dominant life forms, allowing new or previously unexceptional species to multiply into empty ecological niches. If we look back to just prior to the Cambrian explosion (540 Ma), the Earth was populated by multicellular organisms, plants and fungi. Mammals did not appear until about 200 Ma. And up until the K-T impact at 65 Ma, mammals were small creatures, neither numerous nor diverse. With the K-T impact, so many of the dinosaurs were wiped out that there were available ecological niches. Mammals filled this adaptive space and so multiplied, diversified and grew larger in size. The flourishing of the class mammalia allowed many evolutionary experiments, and increasingly social and intelligent apes emerged, leading to humans. Thus until the K-T impact, mammals had no chance to flourish and therefore humans had no chance of emerging. There simply was no opportunity in the history of the Earth until the K-T impact. It is a plausible scenario that the dinosaurs might have died out a few million years later than the 65 Ma impact date by some entirely terrestrial process. There is evidence that the Earth was experiencing a long term global cooling for millions of years before and after the K-T impact (Novacek, note 13, p. 235, figure 5). Indeed, the cooler climate may be one reason why the cold-blooded dinosaurs could not recover from the mass extinction- they were no longer the most fit animals for the environment. But in such a scenario, the dinosaurs would have died out later, mammals would have flourished later, and humans would have arisen later. Since our species homo sapiens has only existed for about 100,000 years, in this latter scenario the species would probably not exist yet. Without the K-T impact, humans might not exist at all, or they might not exist yet. But before this postulated later time arrives, another set of events such as a mass extinction due to an impact might occur to wipe out mammals before humans ever emerge.

Furthermore, limited evidence indicates that the Triassic-Jurassic boundary extinction is consistent with the standard theory that the impacts clear out the large, dominant life forms, allowing other species to fill the empty niches. On the East Coast of North America there are ancient lake basins containing tens of thousands of footprints. Before the Triassic-Jurassic boundary line, the largest of the carnivorous dinosaurs was about the size of a large dog. Other reptiles, such as the 15 foot long rauisuchian (related to both crocodiles and dinosaurs), were the dominant predators, but they disappear at the boundary. Also before the boundary, 20% of all the footprints were from dinosaurs. Within 30,000 years after the boundary, 50% of all the footprints were dinosauran and over this time the size of the average footprint grew by 20%-- indicating an animal up to twice as large. (Olsen, et al, 2001)]

[4] David W. Hughes, "Comets and Asteroids," Contemporary Physics,35 (1994): 75-93.

[5] P. Farinella and D. Vokrouhlicky, "Semimajor Axis Mobility of Asteroidal Fragments," Science 283 (1999): 1507-1510 at 1507. Not covered here is a "seasonal" Yarkovsky effect due to the obliquity of an object (see section 4.1), because it is only operative on bodies up to 100 meters in diameter. [Farinella and Vokrouhlicky, at 1508.]

[6] H.J. Melosh, "Deep Down at Chicxulub," Nature414 (2001): 861-862.

[7] This article assumes 1 year of college science. For a more thorough treatment of photons and electromagnetic radiation, see Raymond A. Serway, Physics for Scientists and Engineers with Modern Physics, 2^nd Edition, (Philadelphia: Saunders Publishing, 1986): chapters 34-35.

[8] F is the force, delta t is the time period, m is the mass and delta v is the change in velocity. The formula for momentum is p=mv. Photons have no mass, but they do have momentum, and the equation is p=h/λ. h is Planck's constant and λ is the photon wavelength. So, the photon-asteroid situation is analogous to the projectile-recoiling cannon situation and the conservation of momentum equation can be represented as: h/λ=(m)(v). We neglect what happens to the photon (the left side of the equation) because we are interested in what happens to the asteroid.

The momentum imparted at any given instant by a single photon (f)(Δt)=(m)(Δv) to an asteroid can be illustrated as:

[9] First, the asteroid emits (radiates) more photons on the day side. This first characteristic is due to the Stefan-Boltzmann law, which says that the energy per second emitted by an object increases with the 4^th power of the temperature: P=(σ)(T⁴) (total power with σ as the Stefan-Boltzmann constant, 5.67 x 10^-8). Note that since the night side surface still has a temperature, it is still emitting photons. Second, on both day and night side most of the photons it emits are in the infrared range. This second characteristic is due to Planck's Radiation Law, which says that in the range of temperatures an asteroid experiences, most of its radiation is emitted in the infrared wavelengths. For Stefan-Boltzmann law, see Serway, Ibid, at 436-437. For Planck's Law, see Serway at 925-926 and figure 40.2.

[10] For the sake of simplicity, the diagram omits showing that solar photons which strike the day side of asteroid are either absorbed or reflected. The force due to the absorption and reflection of solar photons is known as radiation pressure. Pressure is a force per unit area, and can be expressed in units of Newtons per square meter. On large asteroids, the orbital changes caused by radiation pressure "are smaller than those due to the Yarkovsky effect." [[D. Vokrouhlicky and A. Milani, "Direct Solar Radiation Pressure on the Orbits of Small near-Earth Asteroids: Observable Effects?" Astron. Astrophys.362 (2000): 746-755 at 755]. Radiation pressure is why comet dust tails are always pointed away from the sun. Note that the separate comet ion tail is pushed away by the solar wind. To the best knowledge of this author, no articles have yet been published which model radiation pressure for multikilometer sized asteroids in the asteroid belt. Articles have been published (Vockrouhlicky and Milani) which conclude that radiation pressure has a measurable impact on the orbits of up to 900 meter diameter asteroids when they are near 1 AU from the sun. Conceptually, radiation pressure should have less of an effect on larger asteroids. To be appreciably affected by radiation pressure, a particle must have a high surface to mass ratio. The larger the body, the less possible this is, because surface area increases as 4πr², while volume increases as (4/3)πr³, allowing more space to fill with mass.

[11] Because a researcher very kindly shared data on it. D. Vokrouhlicky (e-mail correspondence 12/11/02).

[12] To derive a rigorous estimate, one would need temperature data for a multitude of points on the surface and then integrate over the entire surface, taking into account the differing directions of the force vectors emanating from each point. These types of calculations can be seen on the website of David Vokrouhlicky and Miroslav Broz: http://sirrah.troja.mff.cuni.cz/~mira/mp/Yarko_fam/dora_eos_themis.html.

[13] Derivation of force formula: 1) Preliminaries. P=(σ)(A)(T⁴) (total power with σ as the Stefan-Boltzmann constant, A= πr², the cross-sectional area of the asteroid, T=temperature in Kelvins); E=h/f (energy of a single photon with h=Planck's constant and f=frequency); p= h/λ=hf/c (alternate expressions for momentum of a single photon). Assume for simplicity that all the emitted photons are at the same frequency and therefore have the same momentum. 2) Algebra. Fact: (Total Power)/(energy of a single photon)=number of photons per unit of time: (σ)T⁴πr²)∕ (hf). Fact: number of photons per unit of time multiplied by the momentum of a single photon equals the total momentum at any given time, which is dp/dt=F (force). Putting this together we have: (σ)(T⁴)( πr²) ∕ (hf) x hf/c = [(σ)(T⁴)( πr²) ∕ c] = F because the hf terms cancel.

[14] Formula from, Weissman, McFadden and Johnson, eds., Encyclopedia of the Solar System, (San Diego: Academic Press, 1999) at 821. Note that omitted from the formula is that the entire expression should be multiplied by cos x, where x is the angle of the asteroid's obliquity, that is, the tilt of its rotation axis relative to the plane of the solar system. Derivation of formula: It derives from the formula given in the previous paragraph. Force_{day side} minus Force_{night side} is:(σ)T⁴)( πr²) ∕ c (day side) - (σ)(T⁴)( πr²) ∕ c (night side) = σπr²/c(T⁴_D- T⁴_N). Factor the expression (T⁴_D- T⁴_N) into (T+ ΔT/2)⁴ - (T-ΔT/2)⁴ then multiply those factors and simplify by omitting insignificant terms, to yield 4T⁴ ΔT/T, which when distributed back into the original force formula gives [4(σπr² T⁴) ∕ c ][ΔT/T]. This agrees with our net force formula in the main text except for the 8/3. The 4 becomes an 8/3 because when the asteroid emits photons, some fly off at an angle perpendicular (normal) to the asteroid surface, but many fly off at various angles. The force vectors of these photons emitted at angles partially cancel each other out. For example, assume two photons are emitted from an asteroid and each exerts a recoil force of 1 unit onto the asteroid. One photon is emitted at 45 degrees and another is emitted at -45 degrees (315 degrees). The sum of the two force vectors is not two, but is 1.41. If a photon is emitted at angle y, then the component of the photon force contributing to the acceleration is (cos y), the projection of the force onto the surface normal at the location of emission.

[15] The "asteroid" is Mars' moon, Deimos, which is thought to be an asteroid captured by the planet's gravity. It has the virtue of being relatively well studied so this type of data is available. The mass of Deimos is 1.8 x 10^-15 kg.

[16] Farinella and Vokrouhlicky, supra at figure 1A. See also, W.F. Bottke, jr., et al., "Dynamical Spreading of Asteroid Families by the Yarkovsky Effect," Science 294 (2001): 1693-1695 and Morbidelli A. and Vokrouhlicky D. (2003), "The Yarkovsky-driven origin of near Earth asteroids", Icarus, in press.

[17] P. Farinella and D. Vokrouhlicky, "Semimajor Axis Mobility of Asteroidal Fragments," Science 283 (1999): 1507-1510; Burns, Lamy and Soter, "Radiation Forces on Small Particles in the Solar System," Icarus 40 (1979): 1-48; http://lasp.colorado.edu/~colwell/astr3750-f00/dec12notes.html; http://www.wikipedia.com/wiki/Yarkovsky_effect

[18] Joseph N. Spitale, "Asteroid Hazard Mitigation Using the Yarkovsky Effect," Science 296 (2002): 77.

[19] E.M. Shoemaker, R.F. Wolfe and C.S. Schoemaker, "Asteroid and Comet flux in the nieghborhood of Earth," Geological Society of America Special Paper, 247 (1990): 155-170.

[20] J.K. Wilson, J. Baumbgardner, M. Mendillo, "Three Tails of Comet Hale-Bopp," Geophysical Research Letters vol. 25, no. 3 (1998). See, http://www.bu.edu/csp/imaging_science/planetary/comet2.html.

[21] Planet X: D.P. Whitmire and J.J. Matese, "Periodic comet showers and planet x," Nature 313 (1985): 36-38. Kuiper belt: A concentration of dormant comets beginning near Neptune (30 AU) and stretching beyond the planets (up to 100 AU=.0015 light years).

[22] The Oort Cloud: The solar system is encased in a cloud of billions of dormant comets beyond the Kuiper belt- about .15-1.5 light years from the sun.

[23] David M. Raup, The Nemesis Affair, revised edition, (New York: W.W. Norton & Company, 1999); and see Professor Richard A. Muller's website: http://muller.lbl.gov/pages/lbl-nem.htm.

[24] Raup at 139 and 144.

[25] 28 million year cycle: W. Alvarez and R.A. Muller, Evidence from crater ages for periodic impacts on the Earth, Nature 308 (1984): 718-720; 31 million year cycle: M.R. Rampino and R.B. Stothers, Nature 308 (1984): 709-712.

[26] "Ad hoc" comes from a latin phrase ad hoc negotium meaning, "for this purpose." An ad hoc hypothesis is, "an auxiliary hypothesis introduced for the sole purpose of saving" a theory or hypothesis that is under challenge. The challenge may come from many sources including accepted physical laws which contradict the hypothesis or experimental evidence which contradicts the hypothesis.¹ Stephen Jay Gould gives an archetypical example of an ad hoc hypothesis in his essay, "Continental Drift."² In the early twentieth century, Wegener's theory of continental drift was not generally accepted. A major weakness of the theory was that there was no known mechanism for it. The solution eventually found was plate tectonics, which made continental drift "the passive consequence of our new orthodoxy..." as Gould puts it (at 166). However, at the time, Wegener proposed that gravity was the force causing the motion of continents.

"Physicists responded with derision and showed mathematically that gravitational forces are far too weak to power such a monumental peregrination. So Alexis du Toit, Wegener's South African champion, tried a different tack. He argued for local, radioactive melting of the oceanic floor at continental borders, permitting the continents to glide through." p. 163

Although Gould does not specifically say, we must assume that Du Toit's radioactivity idea had little empirical or theoretical basis. Wegener proposed a theory, it was challenged by being shown to contradict generally accepted scientific knowledge, and du Toit responded with an idea specially generated for the specific purpose of meeting the challenge. Neither 'Planet X' nor 'Nemesis' match Gould's archetype.

1: This definition of "ad hoc hypothesis" is drawn from Ted Lockhart, professor of philosophy at Michigan Technological University, from his course HU3700, Philosophy of Science and from The Swedish National Encyclopedia, Vol. 1, p. 54, which is cited by Dr Lars Tranvik of Lund University on his course website. The outline of a more extensive discussion can be found in footnote 36 of, Imre Lakatos, "History of Science and its Rational Reconstructions", pp. 381-413 in: James H. Fetzer, ed., Foundations of Philosophy of Science: Recent Developments, (New York: Paragon House, 1993), which then cites further sources.

2: "The Validation of Continental Drift," in Stephen Jay Gould, Ever Since Darwin, (New York: W.W. Norton, 1977).

[27] S. Yabushita, "On the periodicity hypothesis of the ages of large impact craters," Mon. Not. R. Astron. Soc. 334(2) (2002): 369-373; L. Jetsu and J. Pelt, "Spurios Periods in the terrestrial impact crater record," Astronomy and Astrophysics, 353 (2000): 409-418; but the following paper finds there is a strong periodicity: M. Matsumoto and H. Kubotani, "A Statistical Test for Correlation Between Crater Formation Rate and Mass Extinctions," Mon. Not. R. Astron. Soc. 282 (4) (1996).

[28] J.J. Matese, P.G. Whitman, D.P. Whitmire, "Cometary Evidence of a Massive Body in the Oort Cloud," Icarus141 (1999): 354-356; J.B. Murray, "Arguments for the presence of a distant undiscovered solar system planet," Mon. Not. R. Astron. Soc. 309 (1999): 31-34.

[29] J. Horner and N.W. Evans, "Biases in Cometary Catalogues and Planet X," Mon. Not. R. Astron. Soc.335(3) (2002): 641-654.

[30] Sanchez, Preston, Jones, Weissman, et al., "Stellar Encounters with the Oort Cloud Based on Hipparcos Data," The Astronomical Journal, 117:1042-1055 (1999)

[31] See G.W. Wetherill, "End Products of Cometary Evolution: Cometary Origin of Earth-Crossing Bodies of Asteroidal Appearance," pp. 537-556 in R.L. Newburn et al., eds., Comets in the Post-Halley Era (Kluwer-Dordrecht, 1991). J.K. Davies et al., "The lightcurve and colors of unusual minor planet 1998 WU24," Icarus 150(2001):69-77 and J.K. Davies, et al., "The lightcurve and colors of unusual minor planet 1996 PW." Icarus 132 (1998):418-430.

[32] P. Nurmi, M.J. Valtonen, J.Q. Zheng, "Period variation of Oort Cloud flux and cometary impacts on the Earth and Jupiter," Mon. Not. R. Astron. Soc. 327 (2001): 1367-1376.

[33] J. Matese and D. Whitmire, "Tidal Imprint of Distant Galactic Matter on the Oort Comet Cloud," The Astrophysical Journal, 472 (1996): L41-L43. The figure for the disk tide is just the difference.

[34] D.E. Morris and R.A. Muller, "Tidal Gravitational Forces: The Infall of "New" Comets and Comet Showers," Icarus 65 (1986): 1-12.

[35] M.J. Valtonen, J.Q. Zheng, J.J. Matese, P.G. Whitman, "Near-Earth Populations of Bodies Coming from the Oort Cloud and Their Impacts with Planets," Earth, Moon and Planets, 71 (1995): 219-223.

[36] Sanchez et al., supra; e-mail communication with P. Weissman, 12/02/02.

[37] While the author did not find their work until after this paper was written, Rampino and Haggerty (1996) mention the galactic tide and stress its significance, noting that "In this case, major events in the history of life (and possibly geophysical changes) may be tied to the dynamics of the Galaxy." M.R. Rampino and B.M. Haggerty, "The 'Shiva Hypothesis': Impacts, Mass Extinctions and the Galaxy," Earth, Moon, and Planets 72 (1996): 441-460.

[38] We are making an assumption to simplify the mathematics. This is justified by the gravitational analogue of Gauss' law, from which Newtonian gravitation can be derived. See, B. Bertotti and P. Farinella, Physics of the Earth and the Solar System, (Dordrecht: Kluwer Academic Publishers, 1990): pp. 1-3.

[39] Material for this explanation is drawn from: J. Matese and D. Whitmire, "Tidal Imprint of Distant Galactic Matter on the Oort Comet Cloud," The Astrophysical Journal, 472 (1996): L41-L43; J.J. Matese, K.A. Innanen, M.J. Valtonen, "Variable Oort Cloud Flux Due to the Galactic Tide," in "Collisional Processes in the Solar System," Astrophysics and Space Science Library, M. Marov, H. Rickman , eds., 261 (2001): 91-102.

[40] The formula for the magnitude of torque is: T= (r)(F) sin θ. See any university level introductory physics textbook, for example, Raymond A. Serway, Physics for Scientists and Engineers with Modern Physics, 2^nd Edition, (Philadelphia: Holt, Rinehart and Winston, 1986): chapter 11.

[41] The rate of change of angular momentum is the torque; the torque supplies the rate of change of angular momentum. To see this, recall that in a linear system, F=ma and P=mv. F=dP/dt. The force is what changes the momentum of an object. Torque is the rotational analogue of force. Therefore, since force is the cause of the change of the (linear) momentum, torque is the cause of change of the angular momentum. See any introductory university level physics text such as Serway, Physics for Scientists and Engineers, supra., chapter 11. I thank Charles R. Greenwell for helping me understand the concept of torque.