English: Summary

SVG image of the HD 40307 system orbits. The orbital parameters are taken from Mayor et al. (2008), arXiv:0806.4587v1 (astro-ph) "The HARPS search for southern extra-solar planets. XIII. A planetary system with 3 Super-Earths (4.2, 6.9, & 9.2 Earth masses)", Table 2.

Assumptions:

1. It is assumed that the system is coplanar. This cannot be constrained by the radial velocity observations used to detect the planets in this system, but our solar system's planets are nearly coplanar, and both the systems of PSR B1257+12 (planets B and C) and Epsilon Eridani (planet b and the circumstellar disk) are coplanar, see Konacki and Wolszczan (2003), The Astrophysical Journal 591, L147–L150 "Masses and Orbital Inclinations of Planets in the PSR B1257+12 System" and Beust et al. (2006), The Astrophysical Journal 132, 2206–2218 "The Extrasolar Planet ɛ Eridani b: Orbit and Mass respectively, so this assumption is at least reasonable.
2. The direction of rotation around the star is assumed to be the same for each planet. Again, this cannot be constrained by the current radial velocity observations, but is a reasonable assumption since this is true for the major planets in our own solar system, and is predicted by current theories of planet formation. Under these first two assumptions, the orientation of the orbits relative to each other is correct as depicted in this diagram, however their actual orientation in 3D space with respect to external reference points (e.g. the Sun, the galactic centre) is unknown, hence such directions are not indicated on the diagram.
3. A further assumption is that the true masses of the planets are small compared to that of the star. If the masses were comparable to that of the star, the actual semimajor axis corresponding to the measured orbital period would be greater. This assumption is supported by probability: if a random orientation of the orbits is assumed, the probability that a planet is more massive than 10 times its minimum mass is 0.5%, dropping to 0.005% for 100 times the mass, and 0.00005% for 1000 times the mass. Since the minimum masses of the planets are so low compared to the mass of the star, the effect on the orbital radius can be neglected.

Interpreting the diagram

The planetary orbits are drawn in orthographic projection, as viewed from directly above the plane of the system so that the orbital direction is anticlockwise. The position of each planet in its orbit is calculated using the time of periastron in the table in the Mayor et al. paper, and are drawn at a time near the end of the radial velocity data (figure 3 in the paper). This time is chosen so that the effects of both uncertainties in the orbital parameters and potential orbital evolution of the system can be treated as negligible.

While the orbits are drawn to the correct scale on the diagram, the star and the planets themselves are not shown to scale.