Barycenter (astronomy)

Two-body problem

Main article: Two-body problem

The barycenter is one of the foci of the elliptical orbit of each body. This is an important concept in the fields of astronomy and astrophysics. In a simple two-body case, the distance from the center of the primary to the barycenter, r, is given by: r_1 = a \cdot \frac{m_2}{m_1 + m_2} = \frac{a}{1 + \frac{m_1}{m_2}} where:

• r is the distance from body 1's center to the barycenter;
• a is the distance between the centers of the two bodies; and
• m and m are the masses of the two bodies.

The semi-major axis of the secondary's orbit, r, is given by .

When the barycenter is located within the more massive body, that body will appear to "wobble" rather than to follow a discernible orbit.

{{anchor|Sun-Jupiter barycenter}} Primary–secondary examples

The following table sets out some examples from the Solar System. Figures are given rounded to three significant figures. The terms "primary" and "secondary" are used to distinguish between involved participants, with the larger being the primary and the smaller being the secondary.

• m is the mass of the primary in Earth masses ()
• m is the mass of the secondary in Earth masses ()
• a (km) is the average orbital distance between the centers of the two bodies
• r (km) is the distance from the center of the primary to the barycenter
• R (km) is the radius of the primary
• a value less than one means the barycenter lies inside the primary

Example with the Sun

If m ≫ m which is true for the Sun and any planet then the ratio approximates to: \frac{a}{R_1} \cdot \frac{m_2}{m_1}.

Hence, the barycenter of the Sun–planet system will lie outside the Sun only if: \begin{align} {a \over R_\odot} \cdot {m_\mathrm{planet} \over M_\odot} & 1 \ \implies ; {a \cdot m_\mathrm{planet}} & {R_\odot \cdot M_\odot} \ &\approx 2.3 \times 10^{11} ; M_\oplus ; \mbox{km} \ &\approx 1530 ; M_\oplus ; \mbox{AU}; \end{align} that is, where the planet is massive and far from the Sun.

If Jupiter had Mercury's orbit (57900000 km), the Sun–Jupiter barycenter would be approximately from the center of the Sun ( ≈ 0.08). But even if the Earth had Eris's orbit (68 AU), the Sun–Earth barycenter would still be within the Sun (just over from the center).

To calculate the actual motion of the Sun, only the motions of the four giant planets (Jupiter, Saturn, Uranus, Neptune) need to be considered. The contributions of all other planets, dwarf planets, etc. are negligible. If the four giant planets were on a straight line on the same side of the Sun, the combined center of mass would lie at about 1.17 solar radii, or just over , above the Sun's surface.

The calculations above are based on the mean distance between the bodies and yield the mean value r. But all celestial orbits are elliptical, and the distance between the bodies varies between the apses, depending on the eccentricity, e. Hence, the position of the barycenter varies too, and it is possible in some systems for the barycenter to be sometimes inside and sometimes outside the more massive body. This occurs where: \frac{1}{1-e} \frac{r_1}{R_1} \frac{1}{1+e}.

The Sun–Jupiter system, with , just fails to qualify: 1.05 0.954.

Relativistic corrections

In classical mechanics (Newtonian gravitation), this definition simplifies calculations and introduces no known problems. In general relativity (Einsteinian gravitation), complications arise because, while it is possible, within reasonable approximations, to define the barycenter, we find that the associated coordinate system does not fully reflect the inequality of clock rates at different locations. Brumberg explains how to set up barycentric coordinates in general relativity.

The coordinate systems involve a world-time, i.e. a global time coordinate that could be set up by telemetry. Individual clocks of similar construction will not agree with this standard, because they are subject to differing gravitational potentials or move at various velocities, so the world-time must be synchronized with some ideal clock that is assumed to be very far from the whole self-gravitating system. This time standard is called Barycentric Coordinate Time (TCB).

Selected barycentric orbital elements

Barycentric osculating orbital elements for some objects in the Solar System are as follows:

For objects at such high eccentricity, barycentric coordinates are more stable than heliocentric coordinates for a given epoch because the barycentric osculating orbit is not as greatly affected by where Jupiter is on its 11.8 year orbit.

References

References

1. {{cite OED2. barycentre
2. (2005). "Encyclopedia of Geology".
3. (2016-08-09). "If You Think Jupiter Orbits the Sun, You're Mistaken".