Discovering a celestial object using a non-parametric algorithm

Abstract

We describe a method that does not use any orbital parameters, to arrive at the position and mass of a new celestial object, using high-precision orbital state vector data of the rest of the objects in the system. As an illustration of this approach, we rediscover Neptune with remarkable accuracy.

Appendices

Appendix A. Details of the iteration method

Referring to eq. (7), let ${\varvec{r}} '_N$ be defined by the radial distance $r'$, the polar angle $\theta '$ with ${\varvec{r}} _i$ along the polar axis and the azimuthal angle $\phi '$ (the spherical polar coordinates with Sun at the centre). Let ${\varvec{r}} ''_N$ be similarly defined by $r''$, $\theta ''$ and $\phi ''$. We carry out a linear stability analysis by writing the Jacobian matrix at ${\varvec{r}} _N$ and calculating its eigenvalues for both the maps.

Now we define two dimensionless quantities $\alpha$ and $\beta$. $\alpha$ is the angle between ${\varvec{r}} _N$ and ${\varvec{r}} _i$, which ranges from 0 to $\pi$ radian. $\beta$ is the ratio $| {\varvec{r}} _N|/| {\varvec{r}} _i|$ which can, in principle, range from 0 to $\infty$. Here are a few interesting results about maps A and B:

• 1. By symmetry, the eigenvalues mentioned above (which also determine stability) for either map can only depend on $\alpha$ and $\beta$.

• 2. Note that ${\varvec{V}}_i$ is in the $i {\odot }N$ plane (plane defined by object i, Sun and Neptune). Hence, again by symmetry, ${\partial r''}/{\partial \phi '}$, ${\partial \theta ''}/{\partial \phi '}$, ${\partial \phi ''}/{\partial r'}$ and ${\partial \phi ''}/{\partial \theta '}$ are zero for both the maps.

• 3. This means that ${\partial \phi ''}/{\partial \phi '}$ is one of the three eigenvalues (called $\lambda _3$) of each of the maps, with the corresponding eigenvector along ${\hat{{\varvec{ \phi }}}}$ (perpendicular to the $i {\odot }N$ plane). This evaluates to $(1+\beta ^2-2\beta \cos \alpha )^{3/2}/\beta ^3$ for map A and $\beta ^3/(1+\beta ^2-2\beta \cos \alpha )^{3/2}$ for map B. Since their product is 1, at least one of the two maps will be unstable for any $\alpha$ and $\beta$.

• 4. It is straightforward, although laborious (hence not shown), to obtain the remaining $2 \times 2$ submatrices of the Jacobians analytically. The determinants (det) and the trace (Tr) for both the maps satisfy det $+$ 1 $=$ Tr. This means that another eigenvalue (called $\lambda _1$) is 1 for both the maps, for any $\alpha$ and $\beta$. This is consistent with the non-uniqueness of the solution of eq. (2) and the existence of a line of fixed points.

• 5. We call the remaining eigenvalue $\lambda _2$. $|\lambda _2|$ and $|\lambda _3|$ are either both less than 1, both greater than 1 or both equal to 1. This means that for generic $\alpha$ and $\beta$ values, exactly one of the maps will be neutral ($\lambda _1=1$, $|\lambda _2|<1$, $|\lambda _3|<1$) and the other map will be unstable ($\lambda _1=1$, $|\lambda _2|>1$, $|\lambda _3|>1$). Figure 2 gives the regimes of $\alpha$ and $\beta$, where maps A and B respectively, are neutral and the other map is unstable. The boundary between the two regimes is given by $\beta =1/(2 \cos \alpha )$. Along this curve, all the eigenvalues of both the maps are unity. As $\beta \rightarrow \infty$, the regimes are decided by whether the angle $\alpha$ is acute or obtuse.

• 6. Both the maps share the same set of eigenvectors. Above, it was noted that ${\hat{{\varvec{ \phi }}}}$ is one of the eigenvectors. The other two will be perpendicular to ${\hat{{\varvec{ \phi }}}}$ (i.e., in the $i {\odot }N$ plane). Let ${\varvec{e}}_1$ be the eigenvector of the neutral map corresponding to $\lambda _1$. The coordinate along ${\varvec{e}}_1$ does not change on iteration. Let $\gamma$ be the angle made by ${\varvec{e}}_1$ with ${\varvec{r}} _i$ in the $i {\odot }N$ plane. $\gamma$ is shown in figure 3 for $\beta$ corresponding to the mean orbital radii of Uranus and Saturn. Note that $\gamma$ does not deviate much from $\alpha$ for a large enough $\beta$. In the limit $\beta \rightarrow \infty$, $\gamma =\alpha$, i.e., ${\varvec{e}}_1$ will be along ${\varvec{r}} _N$.

Fig. 2

In the regimes marked A and B, maps A and B respectively are neutral. $\alpha$ is in radian.

Fig. 3

$\alpha$ is shown as a black solid line. $\beta$ correspond to Uranus (1.56) and Saturn (3.14). The consequent $\gamma$ are shown as a red dot–dashed line and a blue dashed line respectively. $\alpha$ and $\gamma$ are in radian.

Appendix B. Making the initial guess

We need some preliminary analysis to employ this method to locate Neptune. Initially, we model the orbit of Neptune as a circle around the Sun, coplanar with that of Uranus and Saturn. The parameters characterising Neptune’s orbit will then be its radius R and the position on this circular orbit at a given time.

Fig. 4

$V_i(t)=| {\varvec{V}}_i(t)|$ obtained from5 eq. (3) is shown as a function of time for $i=$ Uranus and $i=$ Saturn.

It is easy to discern that the peaks in figure 4 correspond to conjunctions of Neptune with Uranus and Saturn. Pursuing this, we get an estimate for the time of conjunction with Uranus ($T_c$) as around 1822 CE and the duration between the two conjunctions with Saturn as around 36 years. This gives us 160 years, as an estimate for Neptune’s orbital period, which by Kepler’s third law provides $R \approx 30$ AU.

At any given time, an initial guess for the position of Neptune will result in a unique point on the line of fixed points after convergence. For the initial guess, we use a particular choice of angular coordinates and a range of values around R for the radial coordinate. We start the analysis around $T_c$. For the first time step, we use the same angular coordinates as that of Uranus at conjunction. For the subsequent time steps, the position of Neptune obtained after convergence, can be used to make the initial guess for the angular coordinates.