An orbit seen at an angle keeps everything but its focus

A single inverse-square force allows exactly one family of curves, the conic sections, and one number — the eccentricity — chooses among them. That statement is about orbits as they are, in their own plane. No orbit outside the solar system is observed that way. A binary star, a planet imaged beside its star, a star circling the black hole at the Galaxy’s centre: each is seen from a direction set by where the Earth happens to be, and what is recorded is the orbit’s shadow on the sky.

The shadow is a parallel projection — the orbit is so far away that every line of sight to it is effectively parallel — and a parallel projection is a linear map. That single fact decides almost everything about what the shadow looks like, and it decides it in a way that makes the tilt nearly invisible.

The shadow of an ellipse is an ellipse #

A linear map carries straight lines to straight lines, parallel lines to parallel lines and midpoints to midpoints. It carries a circle to an ellipse and an ellipse to another ellipse. The conic sections were defined by Apollonius as cuts through a cone, which is the same thing as saying they are the projections of a circle — and the family is closed under projection: no projection of an ellipse is a parabola or a hyperbola, and no projection of a hyperbola is closed.

So a tilted orbit looks like an orbit. Its apparent shape is an ellipse with some apparent eccentricity, larger than the true one in general, and an apparent semi-major axis, smaller than the true one in general. Nothing in the shape alone says it has been tilted; every ellipse on the sky could be a face-on orbit of that shape, or a tilted orbit of a rounder shape, or a tilted circle.

The algebra makes the closure exact. A Keplerian orbit in its own plane is $x = a(\cos E - e)$, $y = b\sin E$, with $E$ the eccentric anomaly. Rotating it in its plane, tilting the plane and dropping the coordinate along the line of sight gives

$$\mathbf{r}(E) = \mathbf{A}\cos E + \mathbf{B}\sin E + \mathbf{C},$$

with $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ three fixed vectors on the sky. That is the general form of an ellipse with centre $\mathbf{C}$ and conjugate semi-diameters $\mathbf{A}$ and $\mathbf{B}$. Its apparent semi-axes are the singular values of the matrix whose columns are $\mathbf{A}$ and $\mathbf{B}$, and its apparent foci lie along the larger one, at the apparent eccentricity times the apparent semi-major axis from the centre. Four numbers — the components of $\mathbf{A}$ and $\mathbf{B}$, Thiele and Innes’s constants — describe every apparent orbit, and the equation is linear in them, which is why fitting an apparent orbit is a linear problem once the period, eccentricity and time of periastron are fixed.

Equal areas survive the tilt #

The timing survives as well, and for the same reason.

A linear map multiplies every area by its determinant. For a parallel projection from a plane tilted by $i$, that determinant is $\cos i$, the same everywhere. Sectors swept in equal times have equal areas in the orbit’s plane; their shadows have equal areas on the sky, each smaller by $\cos i$. The apparent orbit sweeps equal areas in equal times about the projected position of the star.

That means the second law cannot detect a tilt. A companion seen to move faster near one end of its apparent ellipse and slower near the other, sweeping equal areas about the star in equal intervals, is doing exactly what a face-on Keplerian orbit does. The star sits somewhere inside the ellipse, the motion is consistent with gravity about that point, and the observer has no dynamical evidence that the plane is inclined.

What the second law does constrain is where the star is. Equal areas in equal times hold about the star’s projected position and about no other point. So the timing of the apparent motion locates the star inside the apparent ellipse, even when the star is too faint or too close to its companion to be seen.

The focus does not survive #

Everything that a projection preserves is affine: lines, parallels, ratios along a line, ratios of areas, centres. A focus is not an affine property. It is defined by distances — the sum of distances from the two foci is constant on an ellipse — and a projection distorts distances differently in different directions. The image of the true focus is a point inside the apparent ellipse, on the line through the apparent centre and the image of the true periastron, but it is not, in general, either of the apparent ellipse’s own foci.

The extreme case shows the principle most plainly.

A circle’s focus is its centre. Projected, the centre goes to the apparent centre — midpoints are preserved — and the apparent ellipse’s foci are somewhere else. The apparent orbit is an ellipse of eccentricity 0.87 with its star at the middle. No inverse-square orbit looks like that: in a true orbit of eccentricity 0.87 the star sits near one end. So the moment the star’s position inside the apparent ellipse is known, the apparent ellipse cannot be the true one, and the discrepancy measures the tilt.

For an eccentric orbit the effect is less blatant but the same in kind. The star’s projected position lies on the apparent major axis only in special orientations, and its distance from the apparent centre, as a fraction of the apparent semi-major axis, is not the apparent eccentricity.

The quadratic start matters in practice. An orbit tilted by ten degrees puts its star a few hundredths of an apparent semi-major axis from the apparent focus, which for a binary whose apparent orbit spans an arcsecond is a few hundredths of an arcsecond — measurable now, not in the nineteenth century. Nearly face-on orbits are the hardest to recognise as tilted, and therefore the hardest to correct; fortunately their correction is also the smallest, since the cosine of a small tilt is nearly one.

Three rotations and a shadow #

An orbit’s orientation in space takes three angles to specify: the tilt of its plane, the direction in which that plane crosses the sky, and the direction of periastron within the plane. The three rotations do not commute, and the order in which they are applied is part of the definition of each angle. Projection then removes one dimension, and the question is which of the three it hides.

The direction in which the plane crosses the sky is the easiest: it rotates the whole apparent ellipse on the sky without changing its shape, so it appears directly as the apparent orbit’s orientation. The tilt and the direction of periastron are entangled. Both change the apparent ellipse’s shape, and different combinations of them, with different true eccentricities, produce the same apparent ellipse. An apparent orbit of a given shape could be a face-on orbit of that shape, a tilted orbit with its periastron along the line of nodes and a smaller true eccentricity, or a tilted orbit with its periastron across the line of nodes and a different one. Six numbers fix an orbit. The timing of the apparent motion gives two of them — the period and the moment of periastron — and the remaining five have to come from the apparent ellipse.

That count is the whole content of the deprojection. Five numbers describe any ellipse on the sky: two for its centre, two for its axes, one for its orientation. Measured from an arbitrary point, the centre’s two say nothing about the orbit. Measured from the star, they are exactly the star’s displacement from the apparent centre, and the five then match the five the true orbit still needs — semi-major axis, eccentricity and three angles — and fix them, up to the reflection that swaps the near and far sides of the plane.

Reading the tilt back #

The inversion has a clean geometric statement: the true orbit is the unique ellipse, with the star at one focus, whose projection is the apparent ellipse. Given the apparent ellipse and the star’s position within it, there is exactly one plane and one ellipse in it that satisfy both, up to the reflection that swaps which side of the sky the plane tilts toward — the projection cannot tell whether the near side of the orbit is above or below the sky’s plane. That ambiguity is broken only by radial velocities, which say which way the companion is moving along the line of sight.

The construction was first done in 1827 by Félix Savary for the binary star ξ Ursae Majoris, the first orbit computed for a pair of stars beyond the Sun’s family. Later methods — Kowalsky’s in the nineteenth century, which writes the apparent ellipse as a general conic and extracts the orbital elements from its coefficients, and the linear formulation of Thiele and Innes — automate the same idea. Every one rests on the fact that the focus is the only feature of a Keplerian ellipse that a projection fails to preserve.

Why it matters which ellipse #

The distinction between the apparent and the true orbit is not cosmetic. Three quantities depend on it.

The semi-major axis. The apparent semi-major axis is a projection, shorter than the true one by an amount depending on the tilt and orientation. The third law converts the true semi-major axis and the period into the total mass, and it goes as the cube of the axis: an apparent axis used in place of the true one at a tilt of 60° underestimates the mass by a factor of about two to eight, depending on orientation.

The eccentricity. The apparent eccentricity is not the true one, and it is almost always larger. An orbit’s eccentricity is a clue to its history — tides circularise close binaries, encounters excite eccentricity in wide ones — and a catalogue of apparent eccentricities would describe how the pairs happened to be oriented rather than how they evolved.

The inclination itself. For a spectroscopic binary, whose radial velocities give the masses only multiplied by the sine cubed of the inclination, an inclination from the apparent orbit turns the minimum masses into actual ones. The same is true for planets: a planet detected by its star’s radial velocity has a minimum mass, and an astrometric orbit — the star’s own tiny ellipse on the sky — supplies the inclination that converts it.

What Herschel saw first #

The first evidence that Newtonian gravity operates beyond the solar system came from exactly this kind of shadow. In 1803 William Herschel reported that several pairs of stars he had measured twenty years earlier had changed their relative position angles — the brighter and fainter members had moved around each other. Castor’s two components had turned through a measurable angle. They were not chance alignments of unrelated stars at different distances; they were bound, and moving.

What Herschel had was a few position angles over two decades: arcs of apparent orbits, with no way yet to tell their shape. The first full orbit, Savary’s, came twenty-four years later, and it was a test of whether the apparent motion could be a projected Keplerian ellipse at all. It could. The deprojection succeeded, the star sat at the focus of the true orbit, and the period and semi-major axis obeyed the harmonic law with a total mass that came out of order the Sun’s. Gravity had been weighed in another solar system.

The apparent eccentricity that so often exceeds the true one has an analogue in orbit fitting that is unrelated to projection. A noisy measurement of a circular orbit always returns a positive eccentricity, because noise cannot make an eccentricity negative, and an eccentricity fitted to noisy data cannot be zero. Both effects push catalogues of binary-star eccentricities upward, one through geometry and one through statistics, and both have to be corrected before the eccentricities say anything about how the pairs formed.

The same shadow in three settings #

Visual binaries were the first case and remain the purest. Two stars resolved on the sky, one traced around the other over decades: the apparent ellipse is measured directly, the primary’s position within it is measured directly, and the deprojection is the textbook construction.

The stars at the Galactic centre are the most dramatic. Stars orbiting the black hole at the centre of the Milky Way have been followed for decades with infrared adaptive optics, and their apparent orbits are ellipses with the black hole’s position — itself invisible in infrared at the required precision — inferred from the area law and from the fact that every star’s true orbit must have it at a focus. One of them, S2, completes its orbit in sixteen years on an ellipse of eccentricity 0.88; its apparent orbit is markedly different from its true one, and the deprojection, together with radial velocities, gave the black hole’s mass and distance with no assumption beyond Kepler’s laws.

Astrometric planets are the newest. A star with a planet moves on a small ellipse around their common centre of mass, and the space astrometry missions measure that ellipse at the level of tens of microarcseconds. The star is the only body seen; the focus is the barycentre, an empty point. The deprojection uses the same constraint as for a visual binary — the barycentre must be at a focus of the true orbit — applied to an apparent ellipse a few hundred-thousandths of an arcsecond across.

A planet seen beside its star #

Planets imaged directly beside their stars are the newest visual binaries, and they show the construction under the most difficult conditions. A planet tens of astronomical units from its star takes decades or centuries to orbit, and the images span only a few years — an arc of a few degrees of the apparent ellipse. Separating the planet from the star’s glare is a measurement problem of its own; fitting an orbit to a short arc is another.

A short arc constrains the apparent orbit badly, and the deprojection inherits every uncertainty. The fits are done by sampling: drawing orbits whose projections pass through the measured positions, and accepting each with a weight set by how well it fits. For an arc of a few degrees the accepted orbits span a wide range of eccentricities and tilts, all with the star at their true focus and all projecting through the same points. The four planets around HR 8799, on orbits tilted only moderately to the sky, have been followed for little more than a decade of periods that run to centuries, and their fitted eccentricities and tilts remain broad. On an arc that short, the star’s displacement from the apparent focus is the least well determined quantity of all, because the apparent focus itself is barely defined — and it is the one quantity that carries the inclination.

Edge on, and the limit of the construction #

The construction degrades as the orbit turns edge on. At a tilt of 90° the apparent ellipse collapses to a line segment, traversed back and forth; the star sits somewhere on it; and the area law, which needed area to sweep, becomes the statement that the companion’s position along the line is a known function of time. The orbit can still be fitted, because the motion along the line is the projection of a Keplerian orbit and has a characteristic shape in time, but the information that distinguished the true ellipse from its shadow — the star’s position off the apparent focus — has been folded into one dimension.

Edge-on orbits are, in compensation, the ones that eclipse. A binary seen exactly edge on shows its stars passing in front of each other, and the eclipses give radii and the inclination directly. The two methods divide the sky between them: face-on and moderately tilted orbits are deprojected from their shadows, edge-on ones are read from their eclipses, and the intermediate ones — tilted enough to flatten the shadow, not enough to eclipse — are the hardest of all.

A family with no member that tells #

The deeper statement is about the conics themselves. Apollonius’s family is closed under projection, and closed under it in a way that preserves exactly the properties a timing measurement uses: equal areas, centres, ratios along lines. A family defined by a force law and a family defined by cutting a cone turn out to be the same family, and the overlap means that a projected orbit is never wrong-looking in shape or speed.

The focus is where the force lives. It is the point the inverse-square law singles out, the place the attracting mass must be, and it is the one feature of the ellipse defined by distances rather than by lines. A projection keeps the geometry of the cone and loses the geometry of the force, and the star’s displacement from the apparent focus is the measure of how much was lost. Every true orbit in astronomy outside the solar system has been recovered from that displacement.

Still open: orbits whose stars cannot be seen #

The construction needs the star’s position, and for the newest applications the star is not seen at all. For an astrometric planet, the focus is an empty barycentre; for a dark companion detected by a star’s wobble, the focus is a black hole or a neutron star. In both cases the focus’s position is inferred from the area law alone — the point about which the apparent motion sweeps equal areas — and its uncertainty sets the uncertainty in the inclination and therefore in the companion’s mass.

The space astrometry catalogues now contain tens of thousands of such orbits, most covering only one or two periods. How well the focus can be located from the timing of a partial orbit, and therefore how well the masses of the unseen companions are known, is the question that decides what those catalogues can say about the population of dark companions with a floor under their masses — black holes in binaries, brown dwarfs, giant planets — that no other method reaches.