Anomalous Sun Flyby of 1I/2017 U1 (`Oumuamua)

Abstract

The findings of Micheli et al. (Nature2018, 559, 223–226) that 1I/2017 U1 (`Oumuamua) showed anomalous orbital accelerations have motivated us to apply an impact model of gravity in search for an explanation. A small deviation from the 1/r 1/r potential, where r is the heliocentric distance, is expected for the gravitational interaction of extended bodies as a consequence of this model. This modification of the potential results from an offset of the effective gravitational centre from the geometric centre of a spherically symmetric body. Applied to anomalous Earth flybys, the model accounts for energy gains relative to an exact Kepler orbit and an increased speed of several spacecraft. In addition, the flat rotation profiles of eight disk galaxies could be explained, as well as the anomalous perihelion advances of the inner planets and the asteroid Icarus. The solution in the case of `Oumuamua is also based on the proposal that the offset leads to an approach and flyby trajectory different from a Kepler orbit without postulating cometary activity. As a consequence, an adjustment of the potential and centrifugal orbital energies can be envisaged outside the narrow uncertainty ranges of the published post-perihelion data without a need to re-analyse the original data. The observed anomalous acceleration has been modelled with respect to the orbit solutions JPL 16 and “Pseudo-MPEC” for 1I/`Oumuamua.

1. Introduction

The astronomical body designated as 1I/2017 U1 (‘Oumuamua) is the first object that has been observed during its passage through the Solar System from the interstellar environment. It was detected by Robert Weryk on 19 October 2017 with the Pan-STARRS telescope system in Hawaii [1]. Observations on 30 October 2017 of the lightcurve and its variation indicated a rotation period of ‘Oumuamua of more than five hours [2]. Belton et al. reported in ref. [3] rotation periods near four and nine hours combined with an excited spin state of ‘Oumuamua, cf. also ref. [4,5,6,7]. The findings by Micheli et al. [8] that ‘Oumuamua’s path deviates from a Kepler orbit resulted in a number of publications with different proposals to account for the additional acceleration. Cometary activity and the recoil resulting from outgassing would be a natural explanation, and is, indeed, “the most plausible physical model” [8]. However, even with long equivalent exposure times no cometary coma or tail activity could be detected [2,9]. The inactivity was also confirmed by Spitzer observations on 21 November 2017 [10], and Katz expressed in ref. [11] “... skepticism of the reported non-gravitational acceleration.” Thus important problems remain, because many other observations showed neither cometary activity nor was any meteor activity detected on Earth, cf. [8,12,13,14,15]; in addition, the shape, consistence and origin of ‘Oumuamua are still debated, cf. [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32].

Solar radiation pressure could produce an excess radial acceleration with a1/^r2 dependence, where r is the heliocentric distance, if the object has a large surface and a very low mass according to refs. [23,33]. The authors of the latter reference even suggest as a possibility that a lightsail could be of “artificial origin”. However, McNeil et al. [34] estimated a density of approximately2 000kg^m−3 for ‘Oumuamua, a typical value for asteroids, cf. [35]. A density of this amount is hardly consistent with a lightsail. Radio SETI observations detected no signal [36].

The observation of an interstellar object in the Solar System raises the question with regard to where it might have come from. Potential sources have been indicated by [37,38,39], although the authors of ref. [37] point out that non-gravitational accelerations on the outbound path pose severe difficulties in determining the approach geometry.

In view of the many open questions concerning the nature of ‘Oumuamua, we discuss in the following sections, whether a modification of the1/r gravitational potential and possible consequences could provide an answer to the anomalous Sun flyby of 1I/2017 U1 (‘Oumuamua). Deviations from expected trajectories of artificial spacecraft had previously been deduced from Earth flyby observations, which were also not fully consistent with Newton’s theory of gravity based on a potential exactly proportional to the inverse distance [40,41,42,43,44]. Many studies have been performed to solve this problem, e.g., in refs. [45,46,47,48,49].

We suggested that the interaction between gravitating bodies is affected by mass conglomerations according to an impact model of gravity [50,51]—based on the ideas of Nicolas Fatio de Duillier [52,53,54]. It results in a modification of the1/r potential U for extended bodies, such as the Sun and the Earth. This could qualitatively explain the anomalous energy gain during Earth flybys [55]. Another application of the graviton impact model in the context of large masses could demonstrate a physical process to explain the anomalous rotation curves of disk galaxies [56,57]. The model has also successfully been applied to explain the secular perihelion advances of the inner planets and the asteroid Icarus with the result that an offset of_ρ⊙=(4 400±500)m in the instantaneous direction to an orbiting body is required to explain the observed perihelion advances [58].

In this paper, we will consider our impact model for the gravitational interaction of 1I/2017 U1 (‘Oumuamua) with the Sun. Since the Sun is not a gravitational point source, but has a nominal radius of1_R⊙N=6.957×¹⁰⁸m , cf. [59]. The effective gravitational centre is, therefore, not expected to coincide with the geometric centre – even if a spherical symmetry of the Sun is assumed – but is situated on a sphere with radius_ρ⊙ around the centre, cf. [51]. We had thought that the observed anomalous acceleration of ‘Oumuamua might also be a consequence of this offset. The calculations in the following sections1 show, however, that the effect of the energy gain with reasonable values of_ρ⊙is too small to cause directly the unexpected acceleration, because of the large radial distances involved.

An indirect approach can, however, be pursued to adjust the orbit calculations. It still depends on the anomalous energy gain near perihelion, because it allows us to question the narrow ranges of the published observational uncertainties presented in Table 1 and to define adjusted trajectories outside these ranges which yield the required accelerations with respect to the expected motion of ‘Oumuamua. It is to be noted here that we do not attempt to analyse original data, instead we use the reported orbit characteristics.

2. Results 2.1. Modification of the Gravitational Potential

The reduced mass m of the two-body system Sun–‘Oumuamua can be approximated by the mass_mOuof ‘Oumuamua according to

[13.4]m=_M⊙mOuM⊙+_mOu≈_mOu.

All calculations are, however, performed for a normalized reduced mass ofm=1kgleading to specific quantities, such as, the specific angular momentumMand the specific approach energy_E∞in the next sections. The use of these specific quantities eliminates the unknown mass of the attracted body and simplifies all equations.

The gravitational potential of the system can then be written as a specific quantity and would read under the assumption of the Sun as a point source:

[15.1]U(r)=−εr,

whereε=1.3271244×^1020m3s−2 is the nominal solar mass parameter [59] andr=|r|is the length of the heliocentric position vector. According to CODATA 2018, the gravitational constant is_GN=(6.67430±0.00015)×^{10−11m3kg−1s−2}and gives a solar mass of_M⊙=1.988409×¹⁰³⁰kg. The specific effective potential energy equation then becomes

[15.2]_Ueff(r)=−εr+^M22^r2,

where the last term is the specific centrifugal energy.

Equation (2) has to be modified for bodies with large mass values, e.g., for the Sun, according to our gravitational interaction model. If the distance of the orbiting body from the Sun is always much greater than1_R⊙N, the displacement_ρ⊙ of the effective gravitational centre mentioned in Section 1 can be assumed to be parallel to the radius vector r in the direction of the orbiting body and, consequently, the specific potential energy will be

_Umod=−εr−_ρ⊙=−ε(r+_ρ⊙)(r−_ρ⊙)(r+_ρ⊙)=−ε(r+_ρ⊙)^r2−_ρ⊙2

and can be approximated with_ρ⊙≪rby

_Umod≈−εr−ε_ρ⊙^r2.

The physical process causing the offset_ρ⊙ , described in refs. [50,51], is that multiple interactions of the gravitons with the massive body occur before they can escape.

2.2. Observations of ‘Oumuamua

The published orbit characteristics of ‘Oumuamua considered in this study are compiled as data arcs No. 1 to 4 in Table 1 and Table 2 from the corresponding references. This selection of different solutions out of many more evaluations available in the literature is listed to show the history of the observations and the resulting orbital parameters.

We suggest that a modification of the gravitational potentialU(r) in Equation (2) to_Umod is required as given in Equation (4) for an adequate evaluation of the data arcs. A first attempt is made in Section 2.3 by varying the offset_ρ⊙without changing_E∞ of orbit No. 4 in Table 2. Although it will turn out that the effect with reasonable offset values is very small for ‘Oumuamua, the deviation from an exact1/r potential justifies to scrutinize in Section 2.4 the tight uncertainties given for the orbit parameters in Table 1. These uncertainties determined from data arcs starting more than a month after the perihelion passage might not be representative for the modified potential. An indication of this expectation is that the orbit determinations No. 1 to 4 each exceed the uncertainty limits of e and q of the previous solution. For No. 3, the following statement2, related to the anomalous trajectory, supports this prospect: “The behavior of these accelerations outside the observed data arc from 14 October 2017 to 2 January 2018 can only be assumed. Predictions outside this time interval, particularly prior to October 2017, could be much more uncertain than reported here.” Bailer-Jones et al. in [37] also point out that it is challenging to estimate the inbound leg of the trajectory under these conditions.

From the published eccentricity values e and perihelion distances q of the orbits in Table 1 (rounded to significant digits), the other quantities have been calculated under the assumption of hyperbolic Kepler motions:

The “semi-axis” a of the hyperbola, the semi-latus rectum p and the specific approach energy_E∞with

[15.9]a=qe−1(a);p=q(e+1)(b);_E∞=ε2a(c),

the specific angular momentum and the eccentricity with

[15.4]M=εp(a)e=1+2^M2_E∞^ε2(b)

the approach speed at infinity with

_V∞=2_E∞cf.[18.3(1)]

and the impact parameter with

h=M_V∞cf.[18.3(2)].

2.3. Orbit Modification by Offset_ρ⊙Ou

The laws of conservation of energy and angular momentum determine the motion of a body in a central gravitational configuration:

[14.4]12^drdt2=_E∞−U(r)−^M22^r2=12^r˙2=_E∞+εr−^M22^r2,

wheredr/dt=r˙ is the radial velocity. This equation describes a Kepler orbit with an unmodified Equation (2). There is no radial velocity_r˙periat perihelion, but the highest tangential speed_Vperi. For a perihelion distance q it is:

12_r˙peri2=_E∞+εq−^M22^q2=0(a);12_Vperi2=_E∞+εq=^M22^q2(b);_Vperi=Mq(c).

With the modified potential_Umod of Equation (4) the Equation (10) reads:

12_r˙mod2=_E∞+εr−_ρ⊙Ou−_Mmod22^r2.

It should be noted that the denominator of the potential energy term depends on_ρ⊙Oufor the modified orbit, whereas the one of the centrifugal energy term does not, because the angular momentum is defined about the centre of the solar mass.

A comparison with the Kepler orbit in Equation (10) can best be made at perihelion q, where_r˙peri=0. From

_E∞+εq−^M22^q2=_E∞+εq−_ρ⊙Ou−_Mmod22^q2=0

then follows an exact solution for_Mmodusing a constant_E∞:

_Mmod=q2_E∞+εq−_ρ⊙Ou,

which is a constant of the motion. Since at the start of the data arcs at a radial distance_r1≫_ρ⊙Ou (cf. Equation (17) and Table 3) and beyond the modified orbit can be approximated by a Kepler orbit, we use Equation (7)b with_Mmodto estimate the modified eccentricity_eOU for the orbit calculations in Section 2.4. The quality of the approximation can be checked after an estimate of_ρ⊙Ou has been obtained (as listed in Table 1 and Table 3). The initial conditions at_r1can be defined by requiring_r1=_r1Ouand_r˙1≤_r˙1Ou, where the radial velocity can only be approximated, because a constant_E∞leads to a constant_r˙mod∞ with Equation (12). This equation will be evaluated in Section 2.43 using the specific angular momentum_Mmod obtained in Equation (14) with varying assumptions for_ρ⊙Ou until a reasonable anomalous acceleration fit has been achieved in Figure 1.

As can be seen from Table 3, an offset of4 900m had only a minor effect on the anomalous accelerations and thus the effect is not sufficient to explain directly the anomalous acceleration found in ref. [8] for ‘Oumuamua: “... which corresponds to a formal detection of non-gravitational acceleration with a significance of about30σ.” This detection allows a range of accelerations between_A1/^(r/au)2and_A1/(r/au), “where_A1is a free fit parameter” with a value of “_A1=(4.92±0.16)×¹⁰⁻⁶m^s−2”.

Since we see no reason as to why the solar flyby of ‘Oumuamua the offset valid for the perihelion advances of the inner planets should be outside the range_ρ⊙=(4 400±500)m , we will use its maximum in the next section. The specific potential energy gain at perihelion in Equation (14) would beε_ρ⊙/^q2=444^m2s−2 and the specific centrifugal energy is also increased by that amount4.

2.4. Orbital Parameter Adjustments

As outlined in Section 2.2, the uncertainties in the orbital parameters might be much larger than given in Table 1. In particular, we suspect that the relation between the specific approach energy in Equation (6)c and the specific angular momentum in Equation (7)a is not adequately constrained and, therefore, adjusted quantities outside the narrow limits can later be used in Equation (16). In Table 2, the energy and angular momentum quantities are given in detail together with other supplementary data.

The maxima of_E∞andMhave been obtained from the extreme values of e and q of the orbit solutions No. 3 and 4. Taking the above considerations into account, we argue that we are justified to increase the specific approach and centrifugal energies of the adjusted orbits beyond the maximum1σ values in Table 2 and compare them with orbits No. 3 and 4 in an attempt to model the anomalous accelerations on the data arcs. The details of the adjustment can only be estimated by a trial and error method to be explained below, after the specific approach energy had been increased by a certain amountΔEto

_Eadj=_E∞+ΔE.

Although the variation of the specific angular momentum in the adjusted orbit equation

12_r˙adj2=_Eadj+ε_radj−_ρ⊙−^(M+ΔM)22_radj2≈_E∞+ΔE+ε_radj+ε_ρ⊙radj2−^(M+ΔM)22_radj2=_E∞+ε_radj−^(M+ΔM)2−2ε_ρ⊙−2_radj2ΔE2_radj2=_Eadj+ε_radj−^(M+ΔM)2−2ε_ρ⊙2_radj2=_Eadj+ε_radj−_Madj22_radj2,

is, in principle, not dependent on that of the specific energy, we feel that the observations reflected in the orbit parameters No. 3 and 4 might provide some constraints:

(1) Assuming that the heliocentric distance r of ‘Oumuamua and its radial velocityr˙ could best be established at the beginning of the observations, we require that the initial conditions agree for orbit No. 4 and the corresponding adjusted trajectory. By equating Equations (10) and (16) at_r1=_r1adj

_E∞+ε_r1−^M22_r12=_E∞+ε_r1−^(M+ΔM)2−2ε_ρ⊙−2_r12ΔE2_r12,

the same radial velocity_r˙1=_r˙1adjis valid for both trajectories at this distance. It can be seen that

^(M+ΔM)2=^M2+2ε_ρ⊙+2_r12ΔE

and that the adjusted specific angular momentum as function of the fit parameterΔEis

_Madj=^(M+ΔM)2−2ε_ρ⊙=^M2+2_r12ΔE,

cf. Equation (16). It is important to note that the perihelion distance has to change accordingly. As there are no observations available for q, this cannot be checked and lead to a potential conflict.

(2) For the adjustment of orbit solution No. 3, we refer to a statement by Micheli et al. in the appendix of [8]: METHODS, Section Non-gravitational models. We interpret it to mean that “the non-gravitational acceleration on ‘Oumuamua on October 25 at”r=1.4auwas2.7×¹⁰⁻⁶m^s−2. Assuming that the radial velocity was well defined there by solution No. 3, we use a heliocentric distance_r⋄=1.4au=2.0943702×¹⁰¹¹minstead of_r1 in Equation (19).

With corresponding values for_Eadjand_Madj , the adjusted (or modified) orbit characteristics in Table 1 would be defined by

_eadj≈1+2_EadjMadj2^ε2(a);_aadj=ε2_Eadj(b);_qadj≈_aadj(_eadj−1)(c),

cf. Equations (7)b and (6)a,c. In Section 3 the quality of the approximations will be considered.

In the next step, we have to establish the time dependence of the hyperbolical motions on the trajectories with parametric equations for the time t, the heliocentric distance r and the heliocentric x and y coordinates given in ref. [60] (§15, p. 38)5:

[15.12]t=^a3ε(esinhχ−χ)(a);r=a(ecoshχ−1)(b);x=a(e−coshχ)(c);y=a^e2−1sinhχ(d),

where the parameterχvaries from−∞to+∞ for a complete flyby. The equations have to be applied for orbit characteristics e and a in Table 1, as well as for_eadjand_aadj , cf. Equations (20)a,b, both for orbits No. 3 and 4 under consideration. The same procedure will be used to obtain an estimate of_eOuas a function of_ρ⊙or_ρ⊙Ou.

As listed in Table 3, the time_t1—the start time of the observations in seconds after the perihelion passage—should correspond toχ{0}and_t2, the end of the observations toχ{100}for orbits No. 3 and 4. The formalism allows us to calculate the radial velocitiesr˙ in Equation (10) for both data arcs between_t1and_t2at timest{i}and positionsr{i}for equidistant values of the parametersχ{i}(i=0to100 ). This procedure yields 101 data points and 100 intervals in Equations (22) and (23) (A test with more intervals did not substantially improve the calculations). The same calculations have to be done for the modified and adjusted configurations of Equations (16) and (12) before hundred variations each of the velocity components

Δr˙(χ{i})=r˙(χ{i+1})−r˙(χ{i})andΔ_r˙adj(_χadj{i})=_r˙adj(_χadj{i+1})−_r˙adj(_χadj{i}),

can be determined fori=0to99. The corresponding time intervals are

Δt(χ{i})=t(χ{i+1})−t(χ{i})andΔ_tadj(_χadj{i})=_tadj(_χadj{i+1})−_tadj(_χadj{i}).

The expected accelerations along the trajectories of No. 3 and 4 then are

_A3,4exp{i}=Δr˙(χ{i})/Δt(χ{i}).

The adjusted accelerations are

_A3,4adj{i}=Δ_r˙adj(_χadj{i})/Δ_tadj(_χadj{i})

and_A4Ou, respectively. All accelerations are negative, i.e., decelerations, on the outbound path. Subtraction of the expected greater decelerations from the adjusted values yields the positive accelerations

Δ_A3,4adj{i}=_A3,4adj{i}−_A3,4exp{i}

fori=0to99. By design, it is_t2adj−_t1adj=_t2−_t1for both trajectories and_r1adj=_r1for No. 4. Even for No. 3_r1adj≈_r1is a good approximation. In addition, we have, with a regression coefficient_βr=1.000201, the relation_radj{i}−_r1adj=_βr(r{i}−_r1). Both theΔ_A3,4{i}andΔ_A3,4adj{i} can, therefore, be plotted in Figure 1 over the heliocentric distance r for a comparison with the_A1/(r/au)and_A1/^(r/au)2fits.

The task at hand now is to find specific energy incrementsΔ_E3andΔ_E4or offset values_ρ⊙Ou that lead to additional radial accelerations relative to orbits No. 3 and 4 in reasonable agreement with the range of “non-gravitational acceleration” found by Micheli et al. in ref. [8]. The results with different assumptions forΔEhave been compared by many iterations with this range until acceptable fits withΔ_E3=+2.9×^105m2s−2(for No. 3) andΔ_E4=+4.5×^105m2s−2(for No. 4) were achieved as shown by the functionsΔ_A3,4adj . The relative increase required in Equation (15) thus is≈0.1%.

3. Discussion

Since the expectation outlined in the introduction that an offset of the gravitational centre of the order of several kilometers would provide a direct explanation of ‘Oumuamua’s anomalous radial acceleration could not be substantiated, we pursued an indirect approach and proposed that the deviations from a Kepler orbit permitted us to assume wider uncertainty margins than listed in Table 1 for the orbit solutions No. 3 and 4 until reasonable fits could be produced. As shown in Figure 1, it was possible to obtain appropriate additional acceleration values covering the uncertainty range of ref. [8], but with a steeper decrease as a function of radial distance.

The offset values required to achieve such a fit with constant_E∞ and perihelion q are far outside the range in line with the anomalous perihelion data mentioned in the introduction. The corresponding calculations were, nevertheless, helpful in showing that – even for large offset values – the deviations of the modified orbits from Kepler orbits on the data arcs is so small that the approximation in Equation (20) is very good. In Equation (12), for instance, the correction factors for the potential energy term would be_r1/(_r1−_ρ⊙)=0.99999997and(_r1−_ρ⊙Ou)/_r1=0.99939282at the start of the observations.

The orbits No. 1 and 4 together with the adjusted trajectories during the observation times have been plotted in Figure 2 with the help of Equations (21)c and (21)d. In addition, the trajectories have been traced back to their perihelia by decreasingχ in equidistant steps to zero. It can be seen that orbit No. 4 can barely be distinguished from the adjusted path on this scale. The main difference is indeed the additional acceleration and, consequently, a higher radial velocity at the end of the data arc combined with a greater radial distance. We find from Table 3 for orbit solution No. 3 a velocity_r˙2adjhigher by5.9m^s−1relative to_r˙2and10.1m^s−1for solution No. 4. This corresponds to an increase in the radial distance of_r2adj−_r2=49043kmfor solution No. 4 and21270km for No. 3. The reason for the different results is that alternate initial conditions have been adopted for the adjustments No. 3 and 4. A boost of 40 000 km has been deduced at the end of the observations on 2 January 2018 with NASA’s Hubble Space Telescope (cf. https://www.jpl.nasa.gov/news/news.php?feature=7173, last accessed on 11 February 2019).

The adjusted specific energy_Eadjand the angular momentum_Madj in Table 2 correspond to an increase in_V∞of10.98m^s−1for No. 3 and17.04m^s−1for No. 4 (relative variations 0.0416% and 0.0645%, respectively). The impact parameters h experienced relative increases of≈0.072 1%(No. 3) and≈0.038 1%(No. 4). The adjusted energy and angular momentum quantities are greater than the maximal values_Emaxand_Mmaxallowed within the 1σ -limits listed in Table 1. However, this should not necessarily be seen as a conflict, because of the modified solar gravitational potential in Equation (4). As a consequence of the offset_ρ⊙ , the trajectory of ‘Oumuamua is not an exact Kepler orbit and wider uncertainty ranges can be expected. The arguments presented in Section 2.2 also support this conjecture.

4. Conclusions

The observed anomalous acceleration has been modelled with respect to the orbit solutions JPL 16 and “Pseudo-MPEC” for 1I/‘Oumuamua. with the assumption that the trajectory of ‘Oumuamua is not an exact Kepler orbit allowing us to assume angular momentum and approach energy values_Madjand_Eadjoutside the published uncertainty ranges.

In conclusion, it appears that the observed anomalous acceleration of the interstellar asteroid 1I/2017 U1 (‘Oumuamua) can be modelled without the assumption of any cometary activity.

Figure 1. Shaded areas show the uncertainty ranges of the radial 'non-gravitational acceleration' termsA1/(r/au)(dashed-dotted line) andA1/(r/au)2 (dashed line), cf. [8]. The residual radial accelerationsΔA3adj(r{i})andΔA4adj(r{i}) as differences between orbit solutions No. 3 and 4 and the corresponding adjusted trajectories are plotted as triangles and plus symbols, respectively, cf. Equation (26). IncreasesΔE3andΔE4of the specific energies provided the best overlap of the adjusted graphs with theA1-terms and their uncertainty ranges. The diamond symbol indicates the anomalous radial acceleration2.7×10-6ms-2atr⋄=1.4au (ref. [8]). An offsetρ⊙=100000kmwould result in a residual radial accelerationΔA4Oushown as a solid line.

Figure 2. Hyperbolic orbit solutions for 'Oumuamua (post perihelion) (cf. Table 1, Table 2 and Table 3 ). Perihelion distance q; semi-latus rectum p. orbit No. 1: Observations from 18 to 24 October 2017. The time interval with observations is indicated neart1. Orbit No. 4: Observations from 14 October 2017, 10:32:41 UTC (t1) to 2 January 2018, 11:28:29 UTC (t2). Perihelion on 9 September 2017, 12:10:45 UTC (t0) (Minor Planet Center). The data arc is shown on the adjusted trajectory from timet1tot2. The heliocentric distancer⋄=1.4auis indicated.

No. of Data Arcs		Orbital Parametersb
	Observation Times	e	q/au	q/(1010m)	a/(1011m)
1	18 October 2017–24 October 2017	1.182±0.017	0.2483±0.0068	3.715	2.05
2	14 October 2017–17 November 2017	1.1995±0.0002	0.25534±0.00007	3.819 76	1.914 76
3	14 October 2017–02 January 2018	1.20113±0.00002	0.255912±0.000007	3.828 382 8	1.903 401 0
4	14 October 2017–02 January 2018	1.20117±0.00001	0.255923±0.000003	3.828 555 0	1.903 158 3
Modified orbitc	ρ⊙Ou/m	eOu		q/(1010m)	a/(1011m)
4	4900	1.201 17		3.828 555 0	1.903 158 3
4	1×108	1.201 607 0		3.828 555 0	1.903 158 3
Adjusted orbitsd	ρ⊙/m	eadj		qadj/(1010m)	aadj/(1011m)
3	4900	1.201 706 6		3.836 093 7	1.901 819 0
4	4900	1.201 809 0		3.835 794 5	1.900 705 2

^a Data (shortened to significant values) selected from the following references: No. 1: JPL Small-Body Database; JPL 1; producer Otto Matic; solution date 24 October 2017 (last accessed: 20 February 2019). No. 2: JPL 14; producer Davide Farnocchia; solution date 21 November 2017, cf. https://web.archive.org/web/20171122233227/ https://ssd.jpl.nasa.gov/sbdb.cgi?sstr=2017U1;cad=1 (last accessed: 25 October 2020). No. 3: JPL 16; producer Davide Farnocchia; solution date 26 June 2018; cf. https://ssd.jpl.nasa.gov/sbdb.cgi?sstr=1I (last accessed: 20 October 2020). No. 4: “Pseudo-MPEC” for 1I/‘Oumuamua; created 7 November 2018; IAU Minor Planet Center (MPC), cf. https://www.projectpluto.com/temp/2017u1.htm (last accessed: 10 November 2020).^b Eccentricities: e,_eOuand_eadj ; perihelion distances: q and_qadj;1σ uncertainties of e and q; “semi axes” a and_aadj calculated with Equations (6)a and (20)b; the astronomical unit1au=149597870700m [61].^cAssuming a realistic offset_ρ⊙and an unrealistic large one. Since_E∞ and the flyby distance are assumed to be constant, the “semi axis” a and the perihelion q have not changed either; violating Equation (6)a in the non-Keplerian orbits.^dBy adjusting_Eadjand_Madj ; see Equations (20)a–c.

	Specific Approach	Specific Angular	Semi-Latus	Speed	Impact	Maximum
	Energy	Momentum	Rectum	at Infinity	Parameter	Speed
No. of Data Arcs	E∞/(108m2s−2)	M/(1015m2s−1)	p/(1010m)	V∞/(ms−1)	h/(1011m)	Vperi/(ms−1)
1	3.242	3.279	8.104	25 465	1.288	88 281
2	3.465 51	3.339 14	8.401 53	26 327	1.268 34	87 418
3	3.486 187 1	3.344 155 0	8.426 782 7	26 405.254	1.266 473 3	87 351.636
4	3.486 631 8	3.344 256 5	8.427 294 7	26 406.938	1.266 511 8	87 350.359
See	Equation (6)c	Equation (7)a	Equation (6)b	Equation (8)	Equation (9)	Equation (11)c
	Maximum	1σvaluesa
3	3.486 643 0	3.344 214 5
4	3.486 858 7	3.344 285 3
		Modified	quantitiesb
4	3.486 631 8	3.348 232 9	8.428 973 5	26 406.938	1.267 936 9	87 454.221
		Adjusted	quantitiesc
3	3.489 087 1	3.347 956 6	8.445 952 8	26 416.234	1.267 386 0	87 275.151
4	3.491 131 8	3.347 903 9	8.445 687 0	26 423.973	1.266 994 9	87 280.586

^a Calculated with the help of the uncertainty values in Table 1.^b See Equations (10)–(14).^c See Equations (15)–(19). Relative increases of_E∞maxof 0.070% and 0.123% for orbits No. 3 and 4, respectively, are required to model the anomalous accelerations and 0.111% and 0.108% for^Mmax .

Start of Observationa		Heliocentric Distancer⋄b		End of Observationc
Data arc No. 3d,ewithρ⊙=4.9km
t1=	3 018 127.7 s	t⋄=	4 071 215.1 s	t2=	9 933 475.7 s
χ{0}=	1.008 286 8	χ(23)=	1.157 538 2	χ{100}=	1.657 205 9
t1adj=	3 018 523.2 s	t⋄adj=	4 071 591.1 s	t2adj=	9 933 871.2 s
χadj{0}=	1.008 167 9	χadj{23}=	1.157 479 9	χadj{100}=	1.657 350 3
r1=	1.6468369×1011 m	r⋄=	2.0933441×1011m	r2=	4.3098053×1011m
r1adj=	1.6468869×1011m	r⋄adj=	2.0933173×1011m	r2adj=	4.3100180×1011m
r˙1=	43.550 029 km/s	r˙⋄=	41.351 855 km/s	r˙2=	35.396 199 km/s
r˙1adj=	43.545 919 km/s	r˙⋄adj=	41.351 965 km/s	r˙2adj=	35.402 111 km/s
ΔA1adj=	5.6040×10−6ms−2	ΔA⋄adj=	2.7008×10−6ms−2	ΔA2adj=	0.3714×10−6ms−2
Data arc No. 4e,fwithρ⊙=4.9km
t1=	3 018 115.6 s	t⋄=	4 071 197.8 s	t2=	9 933 463.6 s
χ{0}=	1.008 329 8	χ(23)=	1.157 589 0	χ{100}=	1.657 283 8
t1adj=	3 018 289.0 s	t⋄adj=	4 071 320.9 s	t2adj=	9 933 636.9 s
χadj{0}=	1.008 415 3	χadj{23}=	1.157 763 9	χadj{100}=	1.657 756 8
r1=	1.6468462×1011m	r⋄=	2.0933591×1011m	r2=	4.3098798×1011m
r1adj=	1.6468462×1011m	r⋄adj=	2.0933621×1011m	r2adj=	4.3103702×1011m
r˙1=	43.550 712 km/s	r˙⋄=	41.352 677 km/s	r˙2=	35.397 282 km/s
r˙1adj=	43.550 712 km/s	r˙⋄adj=	41.356 811 km/s	r˙2adj=	35.407 342 km/s
ΔA1adj=	5.3775×10−6ms−2	ΔA⋄adj=	2.6246×10−6ms−2	ΔA2adj=	0.4234×10−6ms−2
Data arc No. 4gwithρ⊙Ou=4.9km
ΔA1Ou=	2.328×10−10ms−2	ΔA⋄Ou=	0.000×10−10ms−2	ΔA2Ou=	0.000×10−10ms−2
Data arc No. 4e,hwithρ⊙Ou=100000km
t1Ou=	3 018 893.4 s	t⋄Ou=	4 072 010.7 s	t2Ou=	9 934 241.4 s
r1=	1.6468462×1011m	r⋄=	2.0933591×1011m	r2=	4.3098798×1011m
r1Ou=	1.6468462×1011m	r⋄Ou=	2.0932806×1011m	r2Ou=	4.3095964×1011m
r˙1=	43.550 712 km/s	r˙⋄=	41.352 677 km/s	r˙2=	35.397 282 km/s
r˙1Ou=	43.539 445 km/s	r˙⋄Ou=	41.345 909 km/s	r˙2Ou=	35.395 830 km/s
ΔA1Ou=	6.3539×10−6ms−2	ΔA⋄Ou=	2.8207×10−6ms−2	ΔA2Ou=	0.2517×10−6ms−2

^aJulian date 2458040.93936 (14 October 2017; 10:32:40.7 UTC, start of observations) and perihelion times of both orbits (see Notes d and e) define times_t1by adjustingχ(0) in Equation (21)a.^bFor orbit No. 3, we assumed_r˙⋄adj≈^r˙⋄at_r⋄adj≈_r⋄according to Constraint (2). The approximations could have been improved by smaller steps of the parameterχ.^cJD 2458120.97811 (2 January 2018; 11:28:28.7 UTC, end of observations) defines times_t2by adjustingχ(100).^dPerihelion on 9 September 2017 12:10:32.5 UTC (JD 2458006.007321) defines time_t0=0sfor No. 3 (JPL 16 for 1I/‘Oumuamua). Adjustment of_χadj{0} by applying Constraint (2) above. e By an appropriate selection of the parameter limit_χadj{100}, the durations of all data arcs become_t2−_t1=_t2adj−_t1adj=_t2Ou−_t1Ou=6915348s.^fPerihelion on 9 September 2017 12:10:45.0 UTC (JD 2458006.007466) defines time_t0=0sfor No. 4 (“Pseudo-MPEC” for 1I/‘Oumuamua). Heliocentric distance_r1adj=_r1equated by adjusting_χadj{0} , cf. Equation (21)b. At_r1it is assumed that_r˙1adj=_r˙1, cf. Constraint (1).^gAnomalous accelerations relevant for orbit No. 4 with_ρ⊙=4.9km.^h The unmodified quantities relevant for orbit No. 4 are not repeated here.

Author Contributions

Both the authors participated equally in developing the physical concept and formulation of the problem. Computation was done at the Max Planck Institute for Solar System Research in Göttingen by K.W. Both authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Acknowledgments

We thank the Max-Planck-Institut für Sonnensystemforschung, Göttingen, and the Indian Institute of Technology (Banaras Hindu University) for organizational support and Peter Czechowsky for constructive criticism of the manuscript. This research has made extensive use of the Astrophysics Data System (ADS), as well as of data provided by the International Astronomical Union's Minor Planet Center and the JPL Small-Body Database. We thank the anonymous referees for their constructive comments and suggestions which improved the manuscript.

Conflicts of Interest

The authors declare no conflict of interest.

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