where K is in m.s-1, P in Earth years, and mp in Jupiter masses. Table 2.3 gives the value of K for various specific instances.
If we continue to neglect the mass of the planet relative to the mass of the star, we may also deduce, from Kepler's Third Law, the semi-major axis of the planet's orbit around the centre of gravity (and thus around the star, because in this approximation, they are assumed to coincide) as a function of the mass of the star and of the orbital period:
As Table 2.3 clearly shows, measurement of the radial velocity of a star to determine the presence of possible planets is a method that is biased towards:
• massive objects (K is proportional to the mass of the planet),
• objects close to their parent star (K is proportional to P-1/3, and therefore greater, the smaller the value of P).
For massive objects, the radial-velocity method is therefore complementary to the astrometric method, which is more sensitive to distant objects. As with astrom-etry, to detect long-period objects the method requires regular observations and an instrumental stability that is monitored over time. We shall return to this point in the discussion of the equipment that enables these measurements to be carried out (Chap. 8).
Unlike the astrometric method, and as indicated earlier, measurement of the radial velocity does not allow us to determine the mass to better than the sine of the angle of inclination. So it does not allow us to observe systems that are viewed face-on. Finally, we should mention that in the case of multiple systems, the radial velocity, taken overall, consists of the sum of the different contributions of the planets, and of their periods of revolution. The detection of multiple systems (such as 55 Cancri, Upsilon Andromedae, etc.), is therefore carried out by analyzing the components of the radial velocity by subtracting the principal component, followed by analysis of the residuals until the measurement noise is reached.
Measurement of the radial velocity of a star is based on the Doppler-Fizeau effect:1 any observer who receives a wave (of whatever nature) emitted at the frequency v by a source in motion, detects it at the frequency v + 8v, where 8v is positive (greater frequency and thus shorter wavelength) if the object is approaching, and negative (lower frequency and greater wavelength) if the object is receding.
A star is a source of electromagnetic waves: every object at a temperature T, has a thermal emission spectrum described, in the black-body model, by the Planck function. The star, as a source of electromagnetic waves may therefore equally be the source of a Doppler-Fizeau effect. The analogies with sound waves ceases here, however, because electromagnetic waves are transverse waves which propagate in a vacuum, whereas sound waves are longitudinal compression waves in a medium (air) that they require to be propagated (sound does not propagate in a vacuum).
In the relativistic expression of the Doppler-Fizeau effect, the wavelength observed is given as a function of the wavelength of the source by the equation:
where Vr is the radial velocity of the source, positive when the object is receding, and c is the velocity of light in vacuum. From this expression we can derive the relative shift in wavelength (or frequency) as:
So, for a hot Jupiter (Vr « 50ms x), the relative shift in wavelength is about 1.5 x 10-7.
Measurement of the radial velocity of a star is therefore carried out by highresolution spectroscopy. The spectrograph resolves the radiation from the star into its different components (corresponding to the 'notes' in our analogy with sound). Measurement of the shift in wavelength is made by observing the overall shift in the heavy-element absorption lines in the spectrum. Given the low amplitude of the shift (Ak/k = 5 x 10-7-10-8, depending on the object), two conditions are required to obtain sufficient accuracy:
• using numerous lines to analyze the shift
• having a wavelength reference to calibrate the spectrum and obtain an absolute measurement of the radial velocity, allowing comparison of results obtained over several years (which is the time required to detect long-period planets).
The first point results in constraints on the spectral resolution of the spectrograph (the value k/Ak where Ak is the spectral range covered by one element of the spectrum). The spectral resolution should be several tens of thousands for the best
1 Fizeau's name should not be dissociated from that of Doppler. In fact, although we owe the observation of the effect to the latter, the true understanding of the phenomena in terms of the actual physics is the work of Fizeau.
modern instruments. There is also a constraint on the choice of targets. Stars that are too young, or are too hot, are difficult to observe (or have a reduced accuracy of measurement), because their spectra do not exhibit sufficient narrow lines for the amount of shift to be measured. The second point requires the use of a reference source, generally a spectral lamp (thorium-argon), or a cell containing a gas, whose absorption spectrum is known extremely accurately (generally a cell containing iodine vapour is used). In either of the solutions, the reference standard should be essentially stable, and thus requires a thermal environment, which itself needs to be stabilized. Measurements are made by exposing the spectrograph's detector to the stellar spectrum and the reference spectrum simultaneously. The best current instruments (such as HARPS on the 3.6-m telescope at La Silla) allow us to obtain a radial velocity to an accuracy better than 1 ms~\ and stable over a period of several years. The principal instruments used for the measurement of radial velocities and their method of operation are described later (Chap. 8).
The detection of radial motion (along the line of sight for an observer on Earth) in a celestial body that is orbited by one or more planets may prove to be simplified if the body periodically emits a signal - electromagnetic waves, for example.
This is particularly the case with pulsars, which are neutron stars that have resulted from the explosion of a supernova, and which have the specific feature of emitting electromagnetic waves in a cone, which sweeps the sky in time with the rapid rotation of the star (with periods between a few milliseconds to several seconds), rather like the beam from a lighthouse. Some of these pulsars are visible from Earth, because the emission cone passes across the Earth at each rotation. They are then detectable as a periodic signal, which is easy to time. If the pulsar exists on its own, the Earth-pulsar distance does not vary and the period of the signal is absolutely constant. If the pulsar has planets, just like the other stars we have been discussing, then it will revolve around the centre of mass of the system, so that during the course of its revolution, the Earth-pulsar distance will increase and decrease, thus increasing or decreasing the travel time of the pulsar's signal. This variation in the travel distance, and thus in the travel time, is indicated by a variation in the period of the pulsar over the course of time.
Assuming, as previously, that the pulsar describes an orbit with a semi-major axis a* about the centre of mass, inclined at angle i relative to the plane of the sky (where i = 0 when the orbit lies in the plane of the sky), the variation in the pulsar's period is given by:
c where c is the propagation velocity of the wave from the pulsar in a vacuum (i.e., the velocity of light).
Table 2.4 Amplitude of the period variations of a pulsar of one solar mass as a function of the planets in orbit
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