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
(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
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
rediscovered thanks to the
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
- 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
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
(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
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,
is used since what is seen is the motion of
the secondary around the primary.
Finally W (denoted O in the applet
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.
This applet is almost totally based on the
IDEA java code generator
(Copyright 1998 Kevin Cooper).
You may find here the java source code, but you
will have to get
the java classes you need to run locally the applet.