The astrometric orbit of a binary

Yes, there is a large fraction of stars which are in fact binaries. Sometimes, when the system is wide enough and the secondary component bright enough, one may see both the primary and secondary components of the system.

However, if the secondary is too much fainter, the binary nature of this system may be detected only by the reflex motion of the primary on the sky. How does it work ? Newton found it a few centuries ago (just think of the planets orbiting the sun or the earth/moon couple). This is one of the way to detect planets around nearby stars.

Imagine that the left image below represents the sky, with the centre being the barycentre (centre of mass) of the system (which itself moves on the sky...). On the right, the two images show the motion in right ascension acosd (alpha*) and declination d (delta) as a function of time (t, in Julian days). Both coordinates are shown in Astronomical Units instead of angle (1 AU is the mean earth-sun distance). The radius of a star is much smaller, so you would need to zoom the images in order to see the stars.

The motion of the primary star is most often smaller than the motion of the secondary. The motion of the photocentre (centre of light), which depends on the magnitude difference between the two components (H₂-H₁) is shown in white. In the case where the two components have equal luminosity (H₂=H₁) and equal mass (M₂=M₁), then the photocentre has no visible motion; on the contrary, when the luminosity of the secondary is too faint, the photocentre is on the primary. The colour of the component motion has been put as a function of the mass (assuming both components on the main sequence).

If you enter a small secondary mass (M₂<0.08), together with its faint magnitude (H₂>14) and a short period (P), you will see how difficult it can be to detect a brown dwarf or a planet through the reflex motion of the primary (see bottom of this page). Now, apart from the masses, you may play with the inclination of the orbit (i) or its excentricity (e), both parameters which will seriously change the shape of the orbit.

The initial parameters which are below are those of HIP 39903, one star in the Hipparcos Catalogue. Its binarity has been detected in 1993, and rediscovered thanks to the Hipparcos data, which allowed to determine the full orbital elements and masses.

This applet provides several controls:

• To change a parameter value, select the parameter whose value you wish to change in the list box at the top (click it with the left mouse button). Then move to the text box to the right of that, and type in the new value (Note: for some obscure reason, for me at least, you should move slowly around this text box before the current value appears). Finally, click the "Set/Redraw" button after each change of a parameter.
• To change the plot area, select the bound you wish to change in the central list box. For example, to change the top endpoint of the vertical axis, you might select the item "Min delta", while to change the bottom endpoint, you would select "Max delta". Then click in the text box to the right of the list box, and type in the new value. Finally, click the "Set/Redraw" button.
• The "Set/Redraw" button sets all parameters, bounds, etc. according to the selections in the list boxes and values in the text boxes.
• The "Clear" button erases old graphs, etc., and draws only the last orbit. It also stops the animation.

Some mathematical background about the way the orbits are behaved:

• First note that what we see is the projection of the true orbit (the orbital plane) onto the sky tangent plane. So one dimension is missing. It can be recovered thanks to the variations of the radial velocity (speed along the line of sight). This method has been the first to allow the discovery of planets orbiting remote stars, since it is independant from distance.
• The semi-major axis (i-e half of the ellipses) is respectively denoted a₁, a₂ and a₀ for the primary, secondary and photocentre orbit. Since the origin is at the barycentre, we have the relation a₁M₁ = a₂M₂. This explains e.g. why the size of the orbit of the primary is smaller when M₂ is smaller.
• Due to the Kepler's third law, the axis, orbital period and masses are related through (a₁+a₂)³ = P²(M₁+M₂). In this equation, the units are easy to remember, when considering the motion of the sun (1) and earth (2): a₁ is negligible, so a₁+a₂ is about 1 AU, P is one year, and M₁ is 1 solar mass, with M₂ negligible compared to M₁. This gives 1³=1²1.
• The orbits are caracterized by 7 parameters and we have already mentionned the semi-major axis a₁ and the period P (usually in days). What about the other parameters?
  The excentricity e describes the elliptical shape of the true orbit, from 0 for a circle, to 1.
  The inclination i is the angle in degrees between the orbital plane and sky plane; the radial velocity is only able to find the mass times sin(i), whereas the astrometric motion would give the exact mass.
  T is the time of the periastron passage: this is the date when the star (or photocentre) is closest to the barycentre in the orbital plane, and it is expressed in Julian Days (JD) minus 2 440 000.0; note that a Julian year is exactly 365.25 days, and that the mean epoch of the Hipparcos mission 1991.25 corresponds to JD 2 448 349.0625.
  The argument of the periastron w₁ (denoted o1 in the applet parameters) is the angle in the orbital plane from the line of nodes to the major axis (the line of nodes is the intersection of the orbital plane and sky plane). The index 1 indicates that it refers to the orbit of the primary, whereas usually for visual observers, w=w₂=w₁+180 deg is used since what is seen is the motion of the secondary around the primary.
  Finally W (denoted O in the applet parameters) is the position angle (from the +d direction) of the line of nodes.

Now, here are the parameters concerning the first extrasolar planet which has been discovered by Mayor and Queloz, from the variation of the radial velocity curve. Since the inclination is unknown, it has been assumed below that it is 30 deg (sin(i)=0.5), so that the mass of the planet is 0.99 Jupiter mass = 0.0009 solar mass. W has been assumed to be 30 deg, and w has no interest for a (almost) circular orbit.