The radial velocity (Doppler wobble) method of planetary detection

4.1 The radial velocity (Doppler wobble) method of planetary detection

Our own Solar System gives us a good insight into this method and its strengths and weaknesses. Astronomers often use, as in this book, the phrase ‘the planets orbit the Sun’. This is not quite true. Imagine a scale model of the Solar System with Sun and planets having appropriate masses and positions in their orbits from the Sun. All the objects are mounted on a fl at, weightless, sheet of support- ing material. By trial and error, one could fi nd a point where the model could be balanced on just one pin. This point is the centre of gravity of the Solar System

Introduction to Astronomy and Cosmology

Figure 4.1 The change in wavelength of a spectral line as the star orbits the barycentre of its solar system.

model. The centre of gravity of the Solar System is called its barycentre , and both the Sun and planets rotate about this position in space (Figure 4.1).

As Jupiter is more massive than all the other planets combined, its mass and position have a major effect on the position of the barycentre, which will thus lie

a distance from the centre of the Sun in the approximate direction of Jupiter. How far might this be? Ignoring all other planets, let us assume that the barycentre lies R km from the centre of the Sun. The mass of the Sun and Jupiter must ‘balance’ about this point so, given R and the semi-major axis of Jupiter’s orbit in km:

M Sun ⫻R⫽M Jupiter ⫻ (777 547 199 – R) The Sun, with a mass of 2 ⫻ 10 30 kg, is approximately 1000 times more massive

than Jupiter. Hence: 1000 R ⫹ R ⫽ 777 547 199

The equatorial radius of the Sun is 695 500 km so, if Jupiter is its only planet, the barycentre of the Solar System would actually lie outside the Sun. When all of the major planets lie on one side of the Sun, as happened in the 1980s – allowing the Voyager spacecraft missions to the outer planets – the barycentre is further from the Sun’s centre and when Jupiter is on the opposite side to the other planets it is nearer

Extra-solar Planets

the Sun’s centre. On average, the barycentre is at a distance of ∼1.25 solar radii from the Sun’s centre, varying between extremes of ∼0.3 and 2 solar radii.

Suppose that we observed the Solar System from a point at a great distance in the plane of the Solar System. We could not see the planets – their refl ected light would be swamped by the light from the Sun – but, at least in principle, we could detect their presence. Due to the Sun’s motion around the barycentre of the Solar System, it would at times be moving towards us and at other times moving away from us. If we could precisely measure the position of the spectral lines in the solar spectrum we could measure the changing Doppler shift and convert that into a velocity of approach or recession. The Solar System as a whole might, of course,

be moving either away or towards us so we would see a cyclical change in velocity about a mean value.

Again, for the sake of simplicity, let us assume that our Solar System has only one planet (Jupiter).

The Sun would be seen to rotate around the barycentre once every 11.86 years, the period of Jupiter’s orbit. Given our calculation of the position of the barycen- tre, we can thus calculate the speed of the Sun in its orbit about the barycentre. The circumference of the Sun’s orbit about the barycentre is:

2 ⫻ π ⫻ 7.77 ⫻ 10 5 km ⫽ 4.9 ⫻ 10 6 km ⫽ 4.9 ⫻ 10 9 m. So, as 11.86 years is 3.74 ⫻ 10 8 s, the orbital speed is:

4.9 ⫻ 10 9 /3.74 ⫻ 10 8 ⫽ 13 m s ⫺1 .

This would mean that the difference between the maximum and minimum veloci- ties would be ∼26 m s ⫺1 .

The current precision in Doppler measurements is of order 2–3 m s ⫺1 , but the hope is that, in time, this might improve to ∼0.5 m s ⫺1 . Very high resolution spec- trometers are used to observe the light from the star whose light is fi rst passed through a cell of gas to provide reference spectral lines to allow the Doppler shift to be measured.

The measurement accuracy of this method would thus be suffi cient to detect the presence of Jupiter in orbit around the Sun. However, in order to be reason- ably sure about any periodicity in the Sun’s motion one would need to observe for at least half a period and preferably one full period. So observations have to be made on a time scale of many years in order to detect planets far from their sun. This is the major reason why few planets in large orbits have yet been detected – the observations have simply not been in progress for a suffi ciently long time.

Introduction to Astronomy and Cosmology

There is one other limitation that you might have realised: should we observe a distant solar system from directly above or directly below, then we would see no Doppler wobble and hence could not detect any of its planets. Unless we have additional information that can tell us the orientation of the orbital plane of a distant solar system, we can only measure the minimum mass of a planet, not its actual mass. If, for example, we later observed such a planet transit across the face of its sun then we would know that the plane of its solar system included the Earth so that the derived mass is the actual mass of the planet, rather than

a lower limit.

A single planet in a circular orbit will give rise to a Doppler curve which is

a simple sine wave. If the orbit of the planet is elliptical, a more complex, but regularly repeating Doppler curve results. In the case of a family of planets, the Doppler curve is complex and will not repeat except on very long timescales. It can, however, still be analysed to identify the individual planets in the system. In a manner similar to the way in which we calculated the orbital motion of the Sun due to Jupiter, one could calculate the Sun’s orbital velocity due to the Earth. This is 0.1 m s ⫺1 , well below the current and predicted future sensitivity of the radial velocity method so other methods are required for the detection of Earth-like planets. As other techniques (discussed below) come to fruition and longer periods of observation are analysed by the radial velocity method, solar systems like our own are beginning to be found but, as yet, we cannot say how common they are.

4.1.1 Pulsar planets

There is one case where Doppler measurements can be made to extreme accuracy. This is when the central object is the remnant of a giant star called a neutron star. These will be discussed in detail in Chapter 7, but all we need to know at this time is that some of these stars (called pulsars) emit regular, very precisely timed pulses and so Doppler shifts can be measured to exceedingly high preci- sion. This, in principle, would easily allow Earth-mass planets to be discovered. However, it is thought that planets would not often survive the massive nuclear explosion – called a supernova – when the pulsar is formed, so such planets are likely to be very rare. However, in 2006 observations made using the Spitzer Space

Telescope showed that the pulsar 4U 0142⫹61 had a circumstellar disc. The disc is thought to have formed from metal-rich debris left over from the supernova explosion that had given rise to the pulsar and is similar to those seen around Sun-like stars, suggesting that planets might be able to form within it. Pulsars emit vast amounts of electromagnetic radiation so such planets would be totally incapable of supporting any form of life!

Extra-solar Planets

Figure 4.2 Planets orbiting the pulsar PSR 1257ⴙ12 at distances of 0.19, 0.36 and 0.46 AU.

This is perhaps how the planets were formed in orbit around the pulsar B1257⫹12. Two planets were initially discovered in 1992 by Aleksander Wolszczan and Dale Frail. Their orbits would both fi t within the orbit of Mercury and they had masses of 4.3 and 3.9 Earth masses (Figure 4.2). Two further planets with masses of just 0.004 and 0.02 Earth masses have since been found in the system and, more recently, a single 2.7 Jupiter-mass planet has been found in orbit around the pulsar B1620–26. It is possible that three other pulsars have one or more planets in orbit around them, but these have yet to be confi rmed.

4.1.2 The discovery of the fi rst planet around a sun-like star

In 1988, Canadian astronomers, Bruce Campbell, G. A. H. Walker, and S. Yang, suggested from Doppler measurements that the star Gamma Cephei might have a planet in orbit about it. The observations were right at the limit of their instruments capabilities and were largely dismissed by the astronomical com- munity. Finally, in 2003, its existence was confi rmed but, unfortunately, this was many years after the fi rst confi rmed discovery of a planet around a main sequence star.

Two American astronomers, Paul Butler and Geoffrey Marcy, were the fi rst to make a serious hunt for extra-solar planets. They began observations in 1987 but, assuming that other planetary systems were similar to our own, did not expect that any planets could be extracted from the data for several years. They would have thus been somewhat shocked when the discovery of a planet orbiting a star called 51 Pegasi was announced by Michael Mayor and Didier Queloz on October 6 1995. The star 51 Pegasi, or 51 Peg for short, lies just to the right of the square of Pegasus and is a Type G5 star, a little cooler than our Sun, with a mass of 1.06 solar masses. Meyer and Queloz were studying the pulsations of stars, which also causes a Doppler shift in the spectral lines as the star ‘breathes’ in and out. With

Introduction to Astronomy and Cosmology

AFOE observations of 51Peg

0.053 P: 4.231 +/–

K: 57.496 +/– 2.871 m/s e: 0.075 +/–

0.001 d w: 132.54 +/– 7.98 To: –1.936 +/– 0.070 (HJD–50000.0)

γ [m/s] 0 –

RV

RMS[residuals]: 8.15 m/s 38 pts fitted reduced xi: 0.85

Orbital Phase

Figure 4.3 Radial velocity measurements of 51 Pegasi made by Korzennik and Contos using the advanced fi breoptic echelle spectrometer on the 1.5 m telescope at the Whipple Observatory near Tucson, Arizona.

a sensitivity of only 15 m s ⫺1 they had not really expected to discover planets but, much to their surprise, they found a periodicity in the motion of the star 51 Peg having a period of 4.23 days and velocity amplitude of 57 m s ⫺1 . The plot is very close to a sinusoid showing that the orbit is very nearly circular (Figure 4.3).

Let us calculate the mass and orbital radius of the planet which is called

51 Pegasi b (Figure 4.4). From the velocity of the star and the period of the orbit we can fi rst calculate the circumference and hence the radius of the star’s motion:

4.23 days is 365 472 s, so the circumference is 57 ⫻ 365 472 m ⫽ 20 831 904 m, giving a radius of 3 315 500 km. This is thus the distance from the centre of the star to the barycentre of the system.

You may remember that we were able to calculate the mass of the Sun given the orbital period and the distance of the Earth from the Sun:

M ⫽ 4π 2 a 3 /GP 2

Extra-solar Planets

Figure 4.4 Artist’s impression of 51 Pegasi b orbiting its sun. Image: Wikipedia Commons.

(Note: this slightly simplifi ed equation assumes the mass of the planet is far less than that of the star.)

As we know the mass of the star 51 Peg (M ⫽ 1.06 ⫻ 2 ⫻ 10 30 kg), and the period P, we can solve this equation for a, the distance of the planet from the star, given the universal constant of gravitation, G.

One AU is 1.496 ⫻ 10 11 m so the planet lies at a distance of 0.052 AU from

51 Peg – this is well within the distance of 0.39 AU at which Mercury orbits our Sun and only about 10 times the radius of the star.

We can now fi nd the mass of the planet by balancing about the barycentre of the system, which we have calculated lies at a distance of 3 315 500 km from the centre of the star. We have:

M planet ⫻ 7.81 ⫻ 10 9 ⫽M star ⫻ 3 315500

Introduction to Astronomy and Cosmology

Giving the mass of the planet: M planet ⫽ 2.12 ⫻ 10 30 ⫻ (3 315 500/7.81 ⫻ 10 9 ) kg

⫽ 9 ⫻ 10 26 kg

Jupiter has a mass of 1.9 ⫻ 10 27 , so the planet has a calculated mass 0.47 that of Jupiter. However, this would only be the mass of the planet if the plane of its orbit included the Earth and will thus be the planet’s minimum mass. One can show that, for random orientations, the mass of a planet will on average be about twice the minimum mass, so the planet in orbit around 51 Peg is likely to be very similar in mass to Jupiter.

When Butler and Marcy learnt about this discovery, they realized that not only could they confi rm its presence from several years observations of 51 Peg in their database – which they did just 6 days later – but that if other massive planets with short periods existed around the stars that they had been observing, they should

be able to rapidly fi nd these as well. This hope was borne out and, to date, they have been the world’s most prolifi c planet hunters. No one had expected that a gas giant would be found so near its star, but many of the planets fi rst discovered were similar in size and separation from their sun. It is not thought that giant planets can form so close to a star, so at some time in their early history it is assumed that they must have migrated inwards through the solar system. In doing so, they would very likely eject smaller (terrestrial type) planets that had formed nearer the star from the solar system and consequently these solar systems are thought unlikely to harbour life.

The Doppler wobble method has proven to be highly successful, but even with

a hoped-for velocity precision of 0.5–1 m s ⫺1 at best, the method will never be able to detect Earth-mass planets, no matter how close they are to their sun.