The “Horseshoe” Orbit of Near-Earth Object 2013 BS45 DANIEL R

Page 20 Astrodynamics The “Horseshoe” Orbit of Near-Earth Object 2013 BS45 DANIEL R. ADAMO, ASTRODYNAMICS CONSULTANT

1. Earth-Based Discovery million km) on 12 February inbound towards the Sun. It 2013. reaches perihelion on 29 April Discovered by the Space- 2013 at 0.92 AU or 92% of watch 1.8 m telescope (see As is typical among NEO Earth’s mean distance from Figure 1) on 20 January 2013, discoveries made in the night the Sun. Figure 2 is a plot of near-Earth object (NEO) 2013 sky prior to closest Earth ap- 2013 BS45, Earth, and Mars as BS45 closely encountered proach with observations of they orbit the Sun during Earth at a range of 0.0126 AU our planet’s night sky, 2013 2013. (4.9 lunar distances or 1.88 BS45 crosses Earth’s orbit

Figure 1. The 1.8 m Space- watch telescope is pictured inside its protective dome at Kitt Peak, Arizona (photograph by Robert S. McMillan).

Figure 2. Orbits of 2013 BS45 (blue), Earth (green), and Mars (red) are plotted during year 2013 in a non-rotating (inertial) Sun-centered (heliocentric) coordinate system. The plot plane coincides with that of Earth’s orbit, the ecliptic, and 2013 BS45’s orbit is inclined to the ecliptic by less than 1°.

Before moving into Earth’s BS45 ephemerides with maxi- planetary radar observations daytime sky circa 9 February mum position uncertainties conducted at Goldstone, CA 2013, about 80 optical obser- equivalent to hundreds of had reduced this uncertainty vations were being processed minutes in heliocentric mo- to the order of 10 minutes. by the Jet Propulsion Labora- tion a century in the past or tory (JPL) to produce 2013 future. During mid-February, (Continued on page 21)

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(Continued from page 20) roundtrip within the time lim- the horizontal axis is Earth its of human exposure to con- departure date, and the verti- Astrodynamics 2. Accessibility finement, galactic cosmic cal axis is roundtrip flight for Human Spaceflight radiation, and microgravity. time in days. Each pixel in a This trade, along with many PCC’s domain is colored ac- Since the orbits of Earth and others, is made by NEO HSF cording to total mission 2013 BS45 are so similar, this Accessible Targets Study change-in-velocity vTOT in NEO should be highly acces- (NHATS, pronounced km/s1. White pixels violate 1 sible for human spaceflight “gnats”) software. The God- one or more NHATS mission In NHATS software, vTOT (HSF) whenever it closely dard Space Flight Center, in viability criteria. Excessive is computed as the sum of approaches Earth. Close ap- impulses required to depart a cooperation with JPL, gener- vTOT or mission duration will proaches are a necessary con- ates and posts NHATS data to generally result from attempt- circular Earth orbit at 400 km dition for HSF accessibility http://neo.jpl.nasa.gov/nhats/ ing to cover an excessive height targeting NEO inter- because of the trade between on a daily basis. roundtrip distance. The PCC cept, achieve NEO rendez- speed and distance in this vous, perform NEO departure for 2013 BS45 appears in Fig- context. Unless distance be- Three-dimensional pork chop ure 3. targeting Earth return, and tween Earth and NEO is charts (PCCs) succinctly sum- ensure Earth’s atmosphere is small, spacecraft speed must marize mission viability under entered at a speed of 12.0 km/ be increased to impractical NHATS criteria. In a PCC s if this value would other- levels in order to complete a plotted by NHATS software, wise be exceeded.

If NHATS mission viability criteria included Earth departure dates circa year 2013, a PCC for 2013 BS45 would be filled with deep blue pixels at relatively short roundtrip flight times for those dates. Unfortunately, about half of these nearly ideal HSF mission opportunities would have been history by the time 2013 BS45 was discovered. As mat- ters stand in early 2013, only about 3 or 4 years would re- main to plan, assemble, and depart Earth before a HSF mission to 2013 BS45 became impractical due to excessive duration and/or excessive

vTOT.

3. Horseshoe Motion With Respect To The Sun-Earth Line

How long will it be before 2013 BS45 again makes close approaches to Earth and

Figure 3. The 2013 BS45 PCC posted 20 February 2013 at the NHATS website shows viable NHATS-viable mission op- mission opportunities on the wane at the earliest NHATS-compliant Earth departure dates portunities resume? Although during year 2015. In accord with the vTOT color legend at right, a NEO with ideal HSF (Continued on page 22)

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(Continued from page 21) accounts for all manner of mula, in which a is the NEO Astrodynamics perturbations to 2013 BS45’s orbit’s heliocentric semi- this NEO’s orbit requires heliocentric motion, chief major axis and  is the Sun’s some refinement before long- among these being Earth’s reduced mass. duration predictions can be gravity. Figure 4 motion made with high confidence, spans one synodic period, the answer appears to be extending from year 1932, “about 80 years”. This inter- when 2013 BS45 last began a val between successive clus- series of close Earth ap- ters of close Earth approaches proaches, through year 2015, With Earth’s heliocentric an- is equivalent to a NEO’s syn- when the current series of gular rate E well determined, odic period. Figure 4 plots close Earth approaches ends. the NEO’s synodic period TS is the time required for the motion of 2013 BS45 in the ecliptic plane with respect to a Kepler’s third law is often difference in angular rate be- rotating Sun-centered coordi- used to compute the heliocen- tween NEO and Earth to ac- nate system in which the Sun- tric angular rate  of a NEO’s (Continued on page 23) Earth line is fixed. This plot orbit using the following for-

2 In this context, “Earth’s vicinity” refers to observations made at Earth’s surface and at contemplated space- based locations ranging out to Sun-Earth libration points about 1.5 million km from Earth along the Sun-Earth line. These libration points are commonly referred to as SEL1 (lying between Earth and the Sun) and SEL2 (lying beyond Earth from the Sun). Figure 4. Heliocentric motion of 2013 BS45, beginning with close Earth approaches in year 1932 and ending with other close approaches in year 2015, is plotted in the ecliptic plane with respect to a fixed Sun-Earth line.

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(Continued from page 22) BS45 during early 2013 as it cally faint they cannot be de- Astrodynamics undergoes an Apollo-to-Aten tected very far from Earth. cumulate a full revolution. transition. Indeed, 2013 BS45 was dis- Therefore, TS = 2 / ( - E). covered only after it had From Table 1’s example, it is closed within 0.044 AU or 6.6 All NEO orbits crossing that evident TS computations ig- million km from Earth. From of Earth are grouped into two noring Earth gravity perturba- its apparent brightness and families. Those Earth-crossers tions on a heliocentric NEO known distance from Earth, with  < E are assigned to orbit cannot produce con- NHATS software estimates the Apollo family, and those sistent or meaningful results 2013 BS45 is 12 to 53 m in with  > E are assigned to at times when those perturba- diameter assuming a reflectiv- the Aten family. A sign con- tions are significant. In such ity range spanning most NE- vention is embedded in the TS instances, a thorough analysis Os of known size. The SEZ formula whereby Aten family of the perturbed orbit must be rules out observing an appre- orbits produce a positive val- conducted from one set of ciable percentage of close ue, and Apollo family orbits close Earth approaches Earth approach points in Fig- produce a negative value. through the next set to infer ure 4. A small NEO with Table 1 presents examples of the actual TS. shorter TS than 2013 BS45’s TS computations for 2013 might have flown past Earth As annotated in Figure 4, the too quickly and evaded dis- plot’s vertical “V” coordinate covery during the brief inter- signifies whether 2013 BS45 val it was close enough to Table 1. When Kepler’s leads (positive V) or trails observe. third law is used to compute (negative V) Earth as they orbit the Sun. Position of tabulated 2013 BS45 synodic A close examination of yearly 2013 BS45 in Figure 4 is anno- period TS values in early “loops” made by 2013 BS45 in 2013, wildly varying results tated for the new year at 10- Figure 4 shows they tend to are obtained as the NEO’s year intervals, beginning with bunch-up when nearest to heliocentric angular rate the initial point at “1932.0”. Earth. This is the graphic transitions from less than Proceeding chronologically manifestation of variations in Earth’s to greater than from this initial point, 2013  previously noted and arises Earth’s. If Earth gravity BS45 trails Earth until the mid from Earth gravity perturba- -1970s when it lies across the perturbations are included tions to 2013 BS45’s heliocen- solar system from our planet when modeling 2013 BS45’s tric orbit. Circa year 1932, and is highly inaccessible for orbit, a TS value near 80 when 2013 BS45 closely trails years is inferred. HSF. Thereafter, 2013 BS45 Earth, these perturbations grows progressively closer to decrease  from slightly more Earth from positions leading 2013 UT TS (years) than E to slightly less than it in orbit about the Sun. 05.0 Jan -201.283 E. In terms of NEO orbit families, 2013 BS transi- The dotted red “v” whose 45 15.0 Jan -209.741 tions from an Aten to an apex coincides with Earth in Apollo and is never able to 25.0 Jan -226.579 Figure 4 denotes a solar ex- overtake Earth. During year clusion zone (SEZ) in which a 2013, similar Earth perturba- 04.0 Feb -283.056 NEO cannot be observed tions are at work to increase 14.0 Feb -3873.767 from Earth’s vicinity because its apparent solar elongation 2013 BS45’s  just before 24.0 Feb +477.737 is less than 40°2. Although Earth would otherwise overtake it. In this scenario, 2013 06.0 Mar +355.733 this zone has infinite extent along Figure 4’s -U axis, its BS45 is transformed from an 16.0 Mar +316.713 boundary is only drawn out to Apollo back to an Aten. Be- cause of the gap surrounding 26.0 Mar +297.532 a geocentric range of 1 AU in Figure 4 because NEOs are Earth in Figure 4, 2013 BS45 05.0 Apr +286.492 typically so small and intrinsi- (Continued on page 24)

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(Continued from page 23) orbit. Figure 5 illustrates 2013 tion in detail from year 2011 Astrodynamics BS45’s Apollo-to-Aten transi- into year 2016. is said to be in a “horseshoe” (Continued on page 25)

Figure 5. The “horseshoe” orbit turnaround is plotted with respect to the Sun-Earth line as 2013 BS45 transitions from the Apollo to the Aten orbit family during years 2011 to 2016. This is a highly magnified segment of Figure 4’s domain.

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(Continued from page 24) relatively short duration and the past, NEOCam would Astrodynamics low vTOT is Earth-NEO dis- have discovered 2013 BS45 as Trajectory markers in the Fig- tance less than 0.1 AU at early as January 2011 at a ure 5 plot are placed at 10-day some point during the mission solar elongation near 106° intervals. Along the trajectory timeline. This criterion ap- and geocentric distance near in the SEZ and all around arc passing closest to Earth at plied to Figure 5 corroborates 0.14 AU. It should be noted Earth’s orbit during reasona- the bottom of this plot, note the best mission opportunities that NEOCam is designed to ble time intervals, consider a only 20 days following 2013 are ending in 2015 just as the observe infrared emissions NEO survey telescope operat- BS45 discovery are available Figure 3 PCC’s time domain quite distinct from reflected ing in interplanetary space in which to observe this NEO opens under NHATS criteria. visible light simulations lead- with perihelion at 0.700 AU from Earth before it drifts into ing to this January 2011 esti- (near the distance of Venus the SEZ. By the time 2013 4. Discovering NEOs Well mate. Nevertheless, a 2013 from the Sun) and aphelion at BS45 departs the SEZ in early In Advance Of Their BS45 discovery two years be- 0.882 AU (near 2013 BS45’s April 2013, 70 days after dis- Close Earth Approaches fore the actual event could perihelion distance). Such an covery, it is likely too far have allowed sufficient time instrument would have a TS of from Earth to detect. It will The NEOCam concept pre- to mount a viable HSF mis- only 2.37 years. Because it hopefully be recovered during sented by JPL/Dr. Amy Main- sion. A “launch on need” ca- remains well inside Earth’s its next close approach during zer in 2009 (ref. link) propos- pability, placing the mission orbit, nearby NEOs observed early 2014, when uncertain- es a NEO survey conducted in a high state of readiness by the telescope in proximity ties in its orbit could then be from SEL1 at solar elonga- before its destination is to Earth’s orbit always have appreciably reduced. tions from 40° to 125°. As- known, might be necessary to solar elongations greater than sume this instrument is capa- visit serendipitously discov- 90°. As such, each observa- It is also useful to consult ble of detecting objects as ered NEOs offering mission tion tends to be of a well- Figure 5 with HSF accessibil- faint as apparent visual mag- opportunities whose Earth illuminated NEO surface near ity in mind. A necessary con- nitude m = +24. With this departure dates are imminent. its maximum possible bright- dition among all NEO round- assumed sensitivity and de- ness from a given distance. trip mission designs boasting ployment sufficiently far in As a means to observe NEOs Figure 6 plots motion of this hypothetical telescope for 10 years using a coordinate system identical to that of Figure 4. This plot begins with the telescope arbitrarily at aphelion near the +U axis on the date 2013 BS45 was discovered. It then extends 10 years into the future. A point near each telescope aphelion is annotated with the corre- sponding date in Figure 6 as 4.2 synodic periods convolve around Earth’s orbit. A telescope in this orbit with m = +24 sensitivity could detect a NEO like 2013 BS45 years or decades before viable HSF mission opportunities would arise.

5. Summary

The orbit of 2013 BS45 serves as a specific example support- ing four important precepts associated with NEOs of high Figure 6. Motion of a notional NEO survey telescope operating between the orbits of Venus HSF accessibility. First, the and Earth is plotted with respect to a fixed Sun-Earth line in the ecliptic plane. The 10-year most accessible NEO orbits interval of this plot spans 4.2 synodic periods for the telescope, ensuring all sectors of inter- tend to be the most Earthlike. planetary space near Earth’s orbit are observed on multiple occasions. (Continued on page 26)

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Astrodynamics (Continued from page 25) tial HSF destination such as synodic periods and have only Such orbits have protracted 2013 BS45 can be observed been observed for the past HSF launch opportunities years or decades in advance decade or two, little is certain several years in duration, but of a close Earth approach. about the long-term dynam- these accessibility seasons These observations will likely ical stability of such orbits. may be separated by intervals leave adequate time to pre- from decades to a century. pare for and utilize the most Acknowledgments practical HSF mission oppor- Second, close NEO approach- tunities. The author gratefully es to Earth associated with acknowledges editorial and HSF mission opportunities are Fourth, some Earthlike NEO technical input from NASA- also the only occasions per- orbits display a horseshoe HQ/Lindley Johnson, NASA- mitting Earthbound observers character in which close ap- HQ/Rob Landis, NASA- to detect small ones ~100 m proaches leading and trailing GSFC/Brent Barbee, and JPL/ in diameter or less. This Earth are achieved with regu- Jon Giorgini. All orbit-related leaves little time to prepare larity, but the Sun-Earth line data appearing in this paper, and dispatch a HSF mission is never crossed. Earth gravity including the simulated 2013 during an accessibility season. perturbations during these BS45 discovery date from ob- close approaches impart turn- servations at SEL1, are trace- Third, by conducting a NEO arounds in the heliocentric able to JPL’s Horizons on- survey from the SEL1 libra- rate at which the NEO is line solar system data and tion point or from interplane- chasing Earth or vice-versa. ephemeris computation ser- tary space between the orbits Because NEOs in horseshoe vice accessible at this link. of Venus and Earth, a poten- orbits possess extremely long Cranium Cruncher Prove this Triangle is Equilateral DOUGLAS YAZELL, EDITOR, FILLING IN FOR DR. STEVEN E. EVERETT

My great friend Jean-Marie first Orion crew capsule Lemaitre showed me this spacecraft will have an ESA brain teaser about two years Service Module. ago. He showed me a quick sketch like the figure at left. MathPages includes animated On May 29, 2013, I was able GIFs. My iPad 1 news aggre- to find the question and an- gator application Flipboard swer thanks to a Google showed me a web site with search which found an unusu- excellent animated GIFs cre- al web site. The web site, ated by PATAKK. Below is a MathPages, seems to be the screen capture image of one creation of author Kevin of those animated GIFs. Brown. The question and answer appear here on his web site, along with excellent history notes. By the way, Jean- Marie teaches mathematics in Hong Kong at the moment. He is no relation to Georges Lemaitre. The fifth Automat- ed Transfer Vehicle (ATV) from the European Space Agency is named after Georges Lemaitre. Based in Above: A web site called MathPages by Kevin Brown (author) part on that excellent ATV includes Morley’s trisection theorem. Image credit: Kevin program success, NASA’s Brown.

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