Is it plausible to expect a close encounter of the Earth with a yet undiscovered astronomical object in the next few years?
aa r X i v : . [ phy s i c s . s p ace - ph ] S e p Is it plausible to expect a close encounter of the Earth with a yetundiscovered astronomical object in the next few years?
Lorenzo Iorio Ministero dell’Istruzione, dell’Universit`a e della Ricerca (M.I.U.R.),Fellow of the Royal Astronomical Society (F.R.A.S.).Viale Unit`a di Italia 68, 70125, Bari (BA), Italy. [email protected]
Received ; accepted 2 –
ABSTRACT
We analytically and numerically investigate the possibility that a still undis-covered body X, moving along an unbound hyperbolic path from outside thesolar system, may penetrate its inner regions in the next few years posing athreat to the Earth. By conservatively using as initial position the lower boundson the present-day distance d X of X dynamically inferred from the gravitationalperturbations induced by it on the orbital motions of the planets of the solarsystem, both the analyses show that, in order to reach the Earth’s orbit in thenext 2 yr, X should move at a highly unrealistic speed v , whatever its mass M X is. For example, by assuming for it a solar ( M X =M ⊙ ) or brown dwarf mass( M X = 80 m Jup ), now at not less than d X = 11 − v would be of the order of 6 −
10% and 3 −
5% of the speed of light c , respectively. By assuming larger present-day distances for X, on the basis ofthe lacking of direct observational evidences of electromagnetic origin for it, itsspeed would be even higher. Instead, the fastest solitary massive objects knownso far, like hypervelocity stars (HVSs) and supernova remnants (SRs), travel at v ≈ . − . c , having acquired so huge velocities in some of the most violentastrophysical phenomena like interactions with supermassive galactic black holesand supernova explosions. It turns out that the orbit of the Earth would not bemacroscopically altered by a close (0 . v/c = 10 − . On the otherhand, this would imply that such a X should be now at just 20 −
30 au, contraryto all direct observational and indirect dynamical evidences.
Subject headings: gravitation; celestial mechanics; planet-star interactions; methods:analytical; method: numerical
1. Introduction
Several free-floating astronomical bodies traveling in the interstellar space in the MilkyWay have been recently detected.In recent years a handful (16) of unbound astrophysical objects lonely wanderingthrough the Milky Way with speeds as large as about v ≈ . c , where c is the speed oflight, have been discovered (Brown et al. 2005; Edelmann et al. 2005; Hirsch et al. 2005;Brown et al. 2006a,b, 2007a,b; Heber et al. 2008; Brown et al. 2009; Tillich et al. 2009;Brown et al. 2010; Irrgang et al. 2010). They are the so-called hypervelocity stars (HVSs),whose existence as a consequence of the Massive Black Hole (MBH) hosted in the centerof the Galaxy (Ghez et al. 2008; Gillessen et al. 2009), was predicted by Hills (1988).Gravitational mechanisms of ejection based on three-body mutual interactions of binarysystems with the MBH, or possibly a pair of MBHs, have been proposed by Hills (1988)and Yu & Tremaine (2003). The consequent rates of HVSs creation would be of the orderof 10 − − − yr − (Perets et al. 2007; Yu & Tremaine 2003). About 10 HVSs may existwithin the Galactic solar circle (Yu & Tremaine 2003). Contrary to those neutron starsexhibiting high proper motions, which are supernova remnants (SRs), known HVSs aremostly B-type main-sequence stars. As an example, HE 0437 − v = 723 km s − = 152 .
517 au yr − = 0 . c (Edelmann et al.2005), is a B star with mass M = 9M ⊙ (Brown et al. 2010). The study by Brown et al.(2010) has yielded the first compelling evidence that these HVSs actually come from thecenter of the Galaxy. All the known HVSs are at about 50 kpc and are unbound withrespect to the Galaxy.Another class of isolated astrophysical objects moving at very fast speeds, not relatedto HVSs, is represented by those neutron stars which are the remnants of asymmetricexplosions of core-collapse supernovæ (SNe) (Burrows 2000). Their extreme speeds arevery likely to be attributed to the kick (Fryer 2004) received in such a kind of peculiardeflagrations . By measuring the displacements of young pulsars from the apparentcenters of their associated SN shells and using the pulsar spin-down periods as ageestimates, Caraveo (1993) and Frail et al. (1994) inferred that pulsars are typically bornwith transverse velocities of 500 km s − , and that velocities v & − may bepossible. At present, the observational record belongs to the radio-quiet neutron star RXJ0822 − − = 331 .
191 au yr − = 0 . c at a distance of 7000 lyr = 4 × au, as measured in 2007 by the Chandra X-rayObservatory (Winkler & Petre 2007). It is thought to have been produced in an asymmetricSN explosion. Indeed, if the explosion of a progenitor star expels the ejecta preferentially in one direc-tion, the compact core must recoil in the opposite direction because of momentum conser-vation. 4 –Moving to isolated substellar objects having smaller velocities by about one order ofmagnitude ( v ≈ − c ), we have the so-called brown dwarfs. They are astrophysical objectsin the range mass M ≈ . − . ⊙ = 41 − m Jup unable to sustain hydrogen fusion intheir cores; as a consequence, it is very difficult to detect them, since most of the energy ofgravitational contraction is radiated away within 10 yr, leaving only a very low residualluminosity. After that their existence was postulated for the first time by Kumar (1963)and Hayashi & Nakano (1963), the first undisputed discovered brown dwarf, and the firstT dwarf, was Gl 229B (Nakajima et al. 1995), with a mass M = 20 − m Jup . After theadvent of large-area surveys with near-infrared (IR) capability in the late 1990’s, hundredsmore brown dwarfs were discovered (Kirkpatrick 2005). Actually, smaller brown dwarf,with
M < m Jup , exist (Lodieu et al. 2007). In particular, in 2005 Luhman et al. (2005)discovered Cha 110913 − M = 8 m Jup ,which is well within the mass range observed for bounded extrasolar planets ( M . m Jup ).An even smaller body, named rho Oph 4450 with M = 2 − m Jup , has been recentlydiscovered by Marsh et al. (2010).Concerning the existence of free-floating planets of smaller mass, Stevenson (1999)noted that, under certain circumstances, Earth-sized solid bodies wandering in theinterstellar space after being ejected during the formation of their parent stellar systemsmay sustain forms of life. Again as a consequence of three-body interactions with Joviangas giants, Debes & Sigurdsson (2007) have recently shown that during planet formation anon-negligible fraction of terrestrial-sized planets with lunar-sized companions will likely beejected into interstellar space with the companion bound to the planet. Debes & Sigurdsson(2007) yield a total number of free-floating binary planets in the Galaxy as large as 7 × .At present, no planets like them have yet been detected. Proposed microlensing surveys ofnext generation will be sensitive to free-floating terrestrial planets (Bennett & Rhie 2002);under certain circumstances, they may be able to yield 10100 detections of Earth-massfree-floating planets (Bennett & Rhie 2002). One to a few detections could be made withall-sky IR surveys (Debes & Sigurdsson 2007).Are there some solitary traveling astronomical objects, still undetected for somereasons, which may hit the Earth over a time scale of a few years? In view of the growingattention that such a possibility may really occur on
21 December 2012 is receiving inlarger portions, also (relatively) educated, of the large public, the present study may alsohave a somewhat pedagogical/educational value contributing, hopefully, to dissipate certainfears too often artificially induced simply for the sake of gain. Mere academic disdainand/or conceit, derision, and hurling insults should not be retained as adequate practices tocounter them. Moreover, the analysis presented here can be repeated in future when other“doomsday” dates will likely pop out. See, e.g., http://en.wikipedia.org/wiki/Nibiru collision on the WEB. 5 –The paper is organized as follows. In Section 2 we present a relatively simplifiedanalytical calculation which, however, grasp the essential features of the situationinvestigated. A more sophisticated numerical analysis is presented in Section 3. It is basedon the numerical integration of the equations of motion by randomly varying the initialconditions. Section 4 summarizes our findings.
2. Analytical calculation
Let us consider a simplified two-body scenario in which a test particle X moves alonga heliocentric hyperbola hurling itself towards the Earth. Its conserved (positive) totalmechanical energy E is (Landau & Lifshitz 1976) E . = 12 µv − αr > , (1)where r and v are the relative X-Sun distance and speed, respectively, µ is the system’sreduced mass µ . = M X M ⊙ M X + M ⊙ , (2)and α . = GM X M ⊙ , (3)where G is the Newtonian constant of gravitation. The semi-major axis a of the hyperbolais determined by its total energy according to (Landau & Lifshitz 1976) a . = α E . (4)The eccentricity e >
1, which, in general, depends on E and on the conserved orbital angularmomentum L , can be fixed by making the simplifying assumption that the periheliondistance of X r (peri) . = a ( e −
1) (5)coincides with, say, the average heliocentric distance of the Earth h r ⊕ i = a ⊕ (cid:18) e ⊕ (cid:19) = 1 . . (6)Thus, e = h r ⊕ i a + 1 . (7) e depends now only on the conserved energy E . The parametric equations for thehyperbola are (Landau & Lifshitz 1976) r = a ( e cosh ξ − ,t = q µa α ( e sinh ξ − ξ ) , (8)where the parameter ξ takes all values from −∞ to + ∞ ; at perihelion ξ = 0. Let us, now,fix r to a given value. It is the heliocentric distance d X at which the putative X shouldbe located at the present epoch. Remaining in the realm of celestial mechanics, d X canbe thought as dynamically constrained by its perturbations of the orbital motions of theknown bound major bodies of the solar system. In particular, upper limits on the tidalparameter of X K X . = GM X d (9)have been recently obtained (Iorio 2009) by using the secular precessions of the longitudesof the perihelia ̟ of the inner planets: for each assumed value of the X’s mass there isa different lower limit for d X . Although, strictly speaking, they have been obtained byassuming X fixed during a planetary orbital revolution, we will use them for the sake ofconcreteness. Of course, if we quite reasonably postulate that X is made of baryonic matteremitting electromagnetic radiation, other, tighter bounds on its present-day distance maybe derived from its electromagnetic direct detectability. The recently launched Wide-fieldInfrared Survey Explorer (WISE) (Wright et al. 2010) will survey the entire sky in themid-IR with far greater sensitivity than any previous all-sky IR surveys (Price 2009) like,e.g., that performed by the Infrared Astronomical Satellite (IRAS) (Beichman 1987).Among the scientific goals of WISE there is also the detection of solitary brown dwarf-likebodies in the neighborhood of the solar system. WISE should be able to reveal the existenceof a body with the mass of Jupiter within
63 kau = 0 . M X = 2 − m Jup would be detectable up to 412 −
618 kau = 2 − −
10 lyr.Moreover, WISE could find a Neptune-sized object out to 700 au. Now, by keeping r = d X it is possible to extract the contemporary value of the parameter ξ corresponding to the Concerning the putative existence of stars and planets made of a particular kind ofnon-baryonic dark matter, i.e. the so-called mirror matter (Lee & Yang 1956), see Foot(1999a,b); Foot & Silagadze (2001). See http://spider.ipac.caltech.edu/staff/tchester/iras/no tenth planet yet.html on theWEB about the alleged discovery of a planet in the remote peripheries of the solar sys-tem a by IRAS. ξ = arccosh (cid:20) e (cid:18) d X a (cid:19)(cid:21) . (10)By substituting eq. (10) into the parametric equation of t of eq. (8), one can plot the timerequired to pass from d X to h r ⊕ i as a function of the alleged velocity of X at the presentepoch: it is sufficient to evaluate E of eq. (1) for r = d X . Thus, from the value of thevelocity required to take a given time interval-typically of a few yr-to reach the Earth’sorbit starting from d X , it is possible to make reasonable guesses about the plausibility ofthe hypothesis that such a putative body X moving towards our planet actually exists outthere.To be more specific, let us assume that X is an object with the mass of the Sun;in this case, Iorio (2009) yields d X = 12 kau as dynamically inferred lower bound of itspresent heliocentric distance. Figure 1 shows that such a Sun-sized body X should travelat a implausibly high speed ( v ≈ . − . c ) to reach our orbit in a few years from now.Recall that the highest recorded speeds of unbound objects of stellar size are 0 . − . c . Β X t H y r L M X = M Sun
Fig. 1.— Time t , in yr, required to a body X with M X =M ⊙ to reach the terrestrial orbitfrom d X = 12 kau as a function of its present day speed β X , in units of c .Note that the situation is even worse if we take a larger value for the limit distance d X in 8 –view of the fact that, after all, a baryonic star should have been easily detected if it wasreally at just 12 kau from us. Indeed, it turns out that by setting, say, d X = 100 kau therequired speed would closely approach c . It may be of interest to note that, by traveling at v = 0 . − . c , a Sun-sized body X would take 300 −
800 yr to reach our orbit if itwas now at 100 kau from us, while 40 −
90 yr would be required if it was at just 12 kau.Incidentally, let us remark that the closest black hole so far discovered, whose distance hasbeen directly measured from its parallax using astrometric VLBI observations, is in theX-ray binary V404 Cyg, at about 2 kpc = 4 × au (Miller-Jones et al. 2009). Anotherclose black hole is V4641 Sgr (Orosz et al. 2001), at about 7 −
12 kpc.Figure 2 depicts the case of a brown dwarf with M = 80 m Jup and d X = 5 . v/c ≈ . − . Β X t H y r L M X = Jup
Fig. 2.— Time t , in yr, required to a body X with M X = 80 m Jup to reach the terrestrialorbit from d X = 5 . β X , in units of c .moreover, for d X = 20 kau it turns out that v/c ≈ . − .
2. No brown dwarfs at allmoving at speeds comparable to those of SNRs and HVSs are known; on the contrary, theirspeeds are of the order of v ≈
100 km s − = 3 × − c (Faherty et al. 2009). Traveling atsuch typical speeds, it would take 1 − d X = 20 kau,and 300 −
900 yr for d X = 5 . d X = 175 au (Iorio 2009),it should travel at v/c = 0 . − .
003 to reach the orbit of our planet in the next fewyr. Given the ejection mechanisms occurring in the planet formation processes which may Β X t H y r L M X = m Ear
Fig. 3.— Time t , in yr, required to a body X with M X = m ⊕ to reach the terrestrial orbitfrom d X = 175 au as a function of its present day speed β X , in units of c .be responsible for such free-floating small planets, their typical velocities should be ofthe order of v ≈ − − = 1 − . × − c for a Jupiter-sized mass ejecting body(Goldreich et al. 2004). Thus, 180 −
300 yr would be required by traveling at such speeds ifan Earth-sized body X was now at d X = 175 au.As we will see in Section 3, the conclusions of such a simplistic analytical two-bodyscenario are also supported by a more sophisticated, numerical analysis.It may be interesting to note that some reflections by M. Brown similar to the reasoningsdeveloped in detail in this Section can be found at http://news.discovery.com/space/mike-brown-planetx-pluto.htmlon the Internet. The case of a body, of unspecified mass, reaching the Earth’s orbit on anunbound trajectory in the next 2 yr starting now from 1 kau is touched. Strictly speaking,the speed of such an unbound X is computed by assuming that it travels uniformly, so thatit is v = 2 . × km s − = 0 . c . According to Iorio (2009), 1 kau is the dynamically 10 –inferred lower limit for a body with M X = m Jup lying perpendicularly to the ecliptic; thespeed required to come here in the next 2 − . . −
1% of c . If wetake d X = 1 . v/c = 0 .
01 to reach 1 au in the next 2 yr. Concerning a Jupiter-sized body X, Brownat http://news.discovery.com/space/mike-brown-planetx-pluto.html puts it at at a fewthousand au; in this case, by setting, say, d X = 2 . v/c = 0 .
3. Numerical calculation
We, first, numerically integrated the equations of motion of an unbound body X in aICRF/J2000.0 heliocentric frame with a coordinate system employing rectangular Cartesiancoordinates along with the ecliptic and mean equinox of reference epoch J2000. In regard tothe initial position chosen for X, we took the predicted coordinates of the Earth at t =21December 2012 retrieved from the HORIZONS WEB interface by NASA/JPL and addedrandomly generated small corrections to them, i.e., x (X)0 = x ⊕ ( t ) + δ x ,y (X)0 = y ⊕ ( t ) + δ y ,z (X)0 = z ⊕ ( t ) + δ z , (11)where δ x , δ y , δ z were randomly generated from a uniform distribution within ± .
001 au.Concerning the initial velocity, we randomly generated it by imposing the conditions v (X)0 > q G ( M ⊙ + M X ) r X0 ,v (X)0 < c, (12)where v p . = r G (M ⊙ + M X ) r X (13)is the limit parabolic velocity; a hyperbola occurs if v > v p . Starting from such sets ofrandomly generated initial conditions, we numerically propagated the trajectory of Xbackward in time over 2 yr, so that t fin represents the present-day epoch. In such a way,by performing several runs, the conclusions of Section 2 turned out to be substantiallyconfirmed in the sense that, in order to avoid finding X at the end of the integration, i.e. atthe present epoch, closer than the dynamically inferred lower limits d X , too high velocitieswould be required. 11 –Then, we made a further numerical analysis in which we used the final state vectors ofX of each of the previous runs backward in time as initial conditions for new runs performed,now, forward in time over 2 yr. In other words, now t corresponds to the present epoch,while t fin = 21 December 2012. In such new runs we also added the Earth, Jupiter andSaturn by modeling their mutual interactions and their attractions on X. Their initialconditions, corresponding to the present epoch, were retrieved from the HORIZONS WEBinterface. The situation remains unaltered: starting today from positions correspondingto the dynamically inferred lower limits d X , all the numerically propagated trajectories ofX reach heliocentric distances of about 1 au in next 2 yr traveling at unrealistically highspeeds, as seen in Section 2. It turns out that, also according to such an analysis, largerinitial distances for X yield even larger speeds for it, just as in Section 2. The inclusionof the major planets of the solar system do not cause noticeable alterations to such apicture. Conversely, from our numerical analysis it turns out that the hypothetical passageof such a fast body X would not distort the orbits of the planets considered, in particularof the Earth. This is clearly depicted by Figure 4 which shows the numerically integratedterrestrial orbit in the next 2 yr in the case of a Sun-sized X body supposed located todayat 11 .
241 kau and moving with v/c = 0 . t fin amounts to ∆ r = 0 . v/c = 2 . × − , with v/v p = 1 .
1. Such a scenario would becatastrophic since in it the Earth would be finally stripped from its orbit and thrown away,as an extension of the time span of the numerical integration to 5 yr shows. Of course, it ishighly unrealistic since it implies the present existence of an undiscovered Sun-sized bodyX at just 26.3 au.
4. Summary and conclusions
We analytically and numerically investigated the possibility-which cyclically gainspopularity for a variety of psychological and/or sociological reasons in extended portionsof the large public, even cultivated-that a yet undiscovered astronomical body X, movingon an unbound trajectory from outside the solar system, may penetrate its inner regionsby closely encountering the Earth in the next few years. For the sake of concreteness wechoose a time span of 2 yr ending at 21 December 2012, familiar to a non-negligible amountof people, but the strategy outlined here can naturally be extended to any temporal intervaland dates in the not unlikely case that in the more or less near future-presumably after2012-other analogous “doomsdays” of astronomical origin will be proposed.As initial positions, we conservatively choose the lower limits d X for the present-daydistance of such a putative X from the bounds dynamically inferred from the magnitudeof the perturbations that it would induce on the orbital motions of the inner planets ofthe solar system. Given that, at present, there are no direct observational evidences of 12 – - - H au L - - y H a u L Earth’s orbit: xy < section Fig. 4.— Section in the { xy } plane of the numerically integrated Earth’s orbit over the next2 yr by assuming for X M X =M ⊙ , d X = 11 .
241 kau, v = 0 . c .electromagnetic origin for the existence of X, tighter constraints on its distance, i.e. largervalues for d X , may well have been adopted. The initial velocities were chosen by allowingfor unbound, hyperbolic trajectories in the field of the Sun. Both analytical and numericalcalculations, performed for different values of the mass of X by randomly varying its initial 13 – - - H au L - - y H a u L Earth’s orbit: xy < section Fig. 5.— Section in the { xy } plane of the numerically integrated Earth’s orbit over the next2 yr by assuming for X M X =M ⊙ , d X = 26 . v = 2 . × − c .conditions and including also the Earth, Jupiter and Saturn, show that, in all cases, Xshould move at unrealistically high velocities to reach heliocentric distances of 1 au in thenext 2 yr. No known astrophysical objects with high speeds, acquired in certain knownphysical processes, move as fast as the putative X should do. In the case of a body withthe mass of the Sun or of a typical brown dwarf ( M X = 80 m Jup ) the speed required to 14 –come close the Earth in the next 2 yr from presently assumed distances of thousands-tenthousands astronomical units would be 6 −
10% and 3 −
5% of c , respectively. Even higherspeeds are involved if we adopt larger values of the initial distance of X relying upon thestill missing direct detection of it from electromagnetic radiation. The fastest Sun-sizedobjects known so far travel at speeds as large as 0 . − .
5% of c , and are produced in some ofthe most violent astrophysical processes known like interactions with supermassive galacticblack holes and supernova deflagrations. Moreover, it turns out that the orbit of the Earthwould not be distorted in a macroscopically noticeable way by the close (0 . .
01% of c . Such a scenario is highly unrealisticbecause, in this case, X should be now at just a few ten astronomical units. 15 – REFERENCES