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

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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) AIAA Houston Section Horizons March / April 2013 Page 20 Page 21 (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 depar- ture 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 mis- sion 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) AIAA Houston Section Horizons March / April 2013 Page 21 Page 22 (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 observa- tions 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. AIAA Houston Section Horizons March / April 2013 Page 22 Page 23 (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.
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