Extrasolar Planets 889
for extrasolar planets, which induce a much smaller reflex
orbit on their host star and produce much smaller velocity
amplitudes.
For a system of two gravitationally bound objectsm 1 and
m 2 in a circular orbit the radial velocity semiamplitudeK 1
ofm 1 can be calculated by using:
K 1 =
(m 2 sini)
(m 1 +m 2 )√
G
(m 1 +m 2 )
am 1 is the more massive object andm 2 is the less massive
secondary companion,idenotes the angle between the or-
bital plane and the plane of the sky,Gis the gravitational
constant, andais the semimajor axis of the orbit. It is im-
mediately clear that for face-on systems (sini=0)K 1 is
zero.
Using Kepler’s famous third law, which relates orbital
separation to orbital period, we can recast this:
K 1 =(
( 2 πG)
P) 1 / 3
(m 2 sini)
(m 1 +m 2 )^2 /^3We are interested in the case of a planet orbiting the star,
wherem 2 m 1 (and thusm 1 +m 2 ≈m 1 ), which simpli-
fies the equation to
K 1 =(
( 2 πG)
P) 1 / 3
(m 2 sini)
m^21 /^3Now we have an expression that relatesm 2 sinito the ob-
servablesK 1 (or simplyKif only the spectrum ofm 1 is
detectable) andP. Using units of years forPandms−^1 for
K,m 2 siniis thus given in Jupiter masses by the following
expression:
m 2 sini=K(
Pm^21) 1 / 328. 4With a good estimate form 1 we thus calculatem 2 sinifor the
unseen companion. Them 2 sinivalue represents a lower
limit to the true mass ofm 2. The siniambiguity is one of
the limitations of the radial velocity technique. However,
them 2 sinivalue is probably close to the real value ofm 2.
Just by assuming a random distribution of orbital planes, we
have a 90% statistical probability thatm 2 is within a factor
of 2.3 of the observedm 2 sini.
Jupiter induces aKof 12.5 m s−^1 in the Sun when ob-
served in the plane of its orbit (sini=1) and Saturn aK
of only 2.8 m s−^1. Figure 2 shows the radial velocity of our
Sun as it would appear to an astronomer in a different plan-
etary system, who happens to observe the Sun from a point
in space that is coplanar to our planetary system. The ob-
served radial velocity signal consists of the superposition of
the signals from the individual planets. In Fig. 2, we see the
FIGURE 2 The radial velocity of our Sun measured from a point
coplanar to the plane of the solar system. The strong signal with a
period of 12 years and a semiamplitude of 12.5 m s−^1 is caused
by Jupiter, while the longer periodic and smaller variation is the
signal caused by Saturn. The radial velocity variations due to the
other planets are negligible.primary 11.86 year period due to Jupiter, with a modulation
due to the orbit of Saturn.
Detection of planets analogous to the two gas giants in
our solar system thus calls for measurement uncertainties
of a few m s−^1 or better over many years to decades. More
massive planets and also planets at smaller orbital separa-
tions produce largerKamplitudes, but the desired velocity
precision is still of the order of several m s−^1.
Over the past years (even decades) two techniques have
been successfully used to attain such a high level of pre-
cision: (1) the gas absorption cell technique and (2) the
simultaneous Thorium–Argon technique in combination
with stabilized spectrometers. In the first method, the
star light is passed through a small glass cell that is filled
with a suitable gas (in most cases iodine vapor), which su-
perimposes its own dense absorption spectrum onto the
stellar spectrum. This reference spectrum not only yields a
simultaneous wavelength calibration but can also be used to
keep track of the imaging properties of the spectrograph.
This allows preventing small changes in the image of the
stellar absorption lines, which are caused by fluctuations
in the light path from the telescope to the detector, from
being misinterpreted as Doppler shifts. In the second tech-
nique, the emission spectrum of a Thorium–Argon lamp
is imaged parallel to the stellar lines on the CCD frame.
Again, this allows a simultaneous wavelength calibration.