Radial Velocity method: Moving around a common center of mass

The Radial Velocity (RV) method is the second-most successful technique to find exoplanets. In contrast to the more successful transit method it has an important advantage: it gives us the mass of the planet. However, it also has at least one major drawback: it is more complicated.

In this post I will try to explain how it works. There are two things you have to realize before you understand why and how the RV method can be used to detect planets.

(1) Although we usually say that a planet orbits a star, this is only half true. Actually, the planet and the star move around a common center of mass. So not only the planet is moving, but the star itself is moving too; however, it is of course not moving as much because it has much more mass.

(2) The light emitted by a moving object is shifted in wavelength/frequency. This is called the Doppler effect and works for all kinds of waves, e.g. sound waves and light. Because the star is moving, the light it emits is shifted to longer wavelengths when moving away from the observer and to shorter wavelengths when moving towards the observer. Light shifted to longer wavelengths is called red-shifted, for shorter wavelengths it is called blue-shifted.

Because the RV method really is about movement, I prepared a little video instead of a static diagram this time. It shows that planet and host star move around a common center of mass indicated by the black cross, and that the light coming towards the observer is shifted in color because of the movement of the star. Only the movement in the direction of the observer is important, which is called the radial movement. This is where the method gets its name from: we only measure the radial component of the velocity.

In the movie the stellar light is not shifted when the planet is exactly behind or in front of the star. In these positions the radial velocity of the star (and the planet too) is zero and no wavelength shift is caused. The higher the velocity of the star in the direction of the observer is, the higher is the wavelength shift.

This also means the RV method works best if we look at a planetary system edge-on. The observer's viewing angle on the system is called inclination i. If i=90° we look at the system directly edge-on and the movement of the star is largest. If i=0° we look at the system from above and the star is not moving in our direction at all - and the RV method does not work anymore.

I tried to illustrate this in a second movie. Now the inclination of the system in respect to the observer has changed and our viewing angle is close to 0°. The radial movement of the star (in direction towards and away from the observer) is much smaller now and, thus, the wavelength shift is smaller too.

This inclination plays an important role, because observers usually do not know its exact value. However, the radial velocity of the star depends on it and, therefore, also the measured mass of the planet depends on the inclination. This is why planetary masses derived with the RV method are described as m sin(i). m is the true mass of the planet and sin(i) is the sine of the inclination. Observers do not measure the true mass, but its "projection" on the angle i. So if we measure a low mass for a planet, this can mean two things: either we really have a low-mass planet with an angle i close to 90°, or we have a higher mass planet with an angle closer to 0° (or 180°). In the end we cannot be sure what kind of planet it is until we know what the inclination is.

The RV method only works as good as we can measure the radial velocity of the star, and this is where the really difficult part begins. If you want to measure the radial velocity of the Sun caused by the gravitational drag of the Earth, you have to have an instrument measuring velocities with a precision of about 10 cm/s. This is tiny. Just compare it to the preferred walking speed of humans, which is already more than a factor 10 higher. The rotation speed of the Sun is roughly 2 km/s. The radius of the Sun is 700000 km, which means you have to measure a change in distance of 10^-10 of the radius of the Sun per second. Or, finally, make yourself aware that Earth orbits the Sun with a speed of about 30 km/s.

Today the best instruments measure radial velocities down to below 1 m/s. Wavelength-stabilized spectrographs observe spectra of the stars and the spectral lines within these spectra can be used to determine their shift due to the Doppler effect. It might be hard to understand how difficult it is to get down to 1 m/s - or even 10 cm/s - if one has never tried to get velocities out of a spectrum. Maybe I can try to show this is one of my next posts in more detail, but think about it that way: the minimum width of a spectral line for the Sun with a rotation velocity of 2 km/s is at least a few km/s. This means you want to measure the position of the line more than a factor 1000 better than its width. The reason why this works at all is that the spectra have hundreds or even thousands of lines.

Exoplanets in the Milky Way: The view from above

Finally my figure of the positions of exoplanets in the Milky Way is finished. Again you see my artist impression of a view from above on the Milky Way, but this time I added the positions of all the known exoplanets for which I could find a distance measurement. The exoplanet data are coming from exoplanet.eu. There are a little bit more than 1000 exoplanets in this map, which means we only have distances for about half of all the exoplanets we know today.

The methods used to detect the planets are indicated by different colors and symbols. Most of the planets in this plot come from the RV method (618) and not from transits (313). At first this might be unexpected, but on average stars observed in RV campaigns are closer to the Sun than transit host-stars because a spectrum needs more light than a brightness measurement. For closer stars the distance is usually easier to determine than for stars far away, for example when using the parallax method.

A quite large number of planets detected by transits orbit stars further away than several thousand light years. This is especially true for those regions in the sky that were observed intensely by transit surveys as for example Kepler. I marked the Kepler field-of-view in the map where several distant planets were found.

The most distant exoplanets, however, were found by microlensing surveys - with the exception of the SWEEPS transit survey. These distant microlensing planets are all located on a line pointing to the center of the Milky Way. Why? This is due to the way the microlensing method works: to see an event we do not only need a planet around a star, we also need a background star which gets 'lensed'. Because the density of stars is highest in the galactic center, the probability to get a lensing event is largest there. Since it is a method which only requires photometric observations, you can see events caused by very distant planetary systems as long as the lensed source is bright enough - or the lensing effect strong enough - to be seen by your telescope.

In case you wonder how scientist get the distances to planetary systems that are so  far away: This would lead too far in this post, but it is not by using the parallax. In these cases distances are usually estimated and, thus, the uncertainties are quite large. Because of the large uncertainty on the distance of the SWEEP exoplanets, one might argue that they possibly are not that far away.

I leave you with a list of names for the most distant planetary systems which are further away from the Sun than 20000 light years. The last two are transiting planets, the rest were all detected in microlensing events.

20961 ly  -  MOA-2010-BLG-353L b
21190 ly  -  OGLE-2005-390L b
22168 ly  -  OGLE-2008-BLG-355L b
22820 ly  -  OGLE-2008-BLG-092L b
23961 ly  -  MOA-2011-BLG-262L b
24058 ly  -  MOA-2011-BLG-028L b
25102 ly  -  MOA-2011-BLG-293L b
25232 ly  -  MOA-2011-BLG-322L b
26732 ly  -  KMT-2015-1 b
27710 ly  -  SWEEPS-4
27710 ly  -  SWEEPS-11

What Hipparcos saw and Gaia will see

In the last post I wrote about the Hipparcos mission and I would like to follow up with a few more nice plots. Hipparcos used the geometric parallax to measure the distances of stars in a rather limited volume around the Sun. Although it measured more than 100000 distances, this covers only a tiny fraction of stars in our Milky Way. Whether Hipparcos can measure the distance to a star depends mainly on two things: The star has to be bright enough to be seen and it has to be close enough to move in the sky by a parallax at least as large as the measurement precision of the instrument.

The latter is slightly better than one milli-arcsecond for Hipparcos and means that only stars that are not much further away than several thousand light years can be measured in distance. The first criterium is called the limiting magnitude, which is about 12 for Hipparcos. It means that stars fainter than an apparent magnitude of 12 are not bright enough to determine the distance. The apparent magnitude depends on the intrinsic brightness of the star (the absolute magnitude) and on the distance - if a star is further away it appears to be fainter. If a star is far away but very bright, it can still be seen by Hipparcos, although the distance cannot be measured if it is so far away that its parallax is smaller than the measurement precision of Hipparcos.

The picture on top shows you again my artist impression of the Milky Way in inverted colors. This time I try to show how far Hipparcos could see for a specific type of star which is called the spectral type. A star with a spectral type M is cooler than the Sun and, therefore, its absolute brightness is lower. The hottest and most luminous stars are O stars. The Sun is a G2 star which has a (surface) temperature of slightly below 6000° Celsius. Unsurprisingly, more luminous stars like O stars can be seen in a much larger distance than cooler stars like F or even M stars. The distance to which a G2 star can be seen by Hipparcos is so small, its smaller than the size of the cross marking the position of the Sun in the picture. But O stars are so very bright, they can be seen throughout the entire galaxy.

In the upper left corner of the plot on top you find the color code for the type of the star and the distance to which this type of star can be seen by Hipparcos. Keep in mind that this does not necessary mean that the distance can be measured just because Hipparcos would be able to see the star.

The region close to the Sun is hard to see in the plot, so I prepared some more pictures to show what is going on there. On the left you see a histogram of the number of stars for a certain distance from the Sun. It shows that within a radius of 100 light years Hipparcos saw 2466 stars, in a radius of 20 light years 'only' 75 stars. The closest stars to the Sun are between 4 and 5 light years away in the Alpha Centauri system: Proxima Centauri, alpha Centauri A and alpha Centauri B.

For the first 400 to 500 light years the number of stars Hipparcos saw increases, then the numbers start to go down. The larger the distance gets, the larger the volume of the shell of the sphere gets in which we are looking for stars. And for the first few hundred light years this is close enough to see more and more stars. However, stars with a low absolute magnitude like M stars get 'invisible' for Hipparcos after a distance of about 120 light years. The further we go away, the more stars become undetectable by Hipparcos. At about 400 to 500 light years the increasing volume of the shell is counter-balanced by the quickly decreasing number of stars that still can be seen, and the absolute numbers start to go down. In a distance of about 4300 light years Hipparcos does not even see A stars anymore, which is where the mission provides virtually no distance measurements anymore. In the cumulative distribution you can see that in a distance of about 500 light years about 50 % of all the stars are located that Hipparcos could measure distances for.

So what ca we do to see more stars and measure their distances? To see more stars we need a better telescope, which practically means a larger telescope. To get larger distances we need a better measurement precision. This is what Gaia is supposed to do. Gaia will have a limiting magnitude of about 20 and will detect stars that are 1600 times fainter than what Hipparcos could see. The precision to measure the parallax will be better than 10 micro-arcseconds, which is more than a 100 times better than Hipparcos; distances of 300000 light years, which is three times the assumed diameter of the Milky Way, should be possible for very bright stars.

The plot at the bottom shows what Gaia will be able to see in terms of brightness. Hipparcos could see a G2 star only in the close neighborhood of the Sun, Gaia will see G stars in a radius of more than 40000 light years - larger than the distance from the Sun to the center of the Milky Way. And stars as luminous as F stars will be visible virtually all over our entire Galaxy.

This way it is assumed that Gaia will see about 1 % of all stars in the Milky Way. This is more than 1 billion stars! However, you still might think: Why 'only' one percent if it can look so 'far'? Well, this is because more than 70 % of the stars in our Milky Way are M stars - and M stars cannot be seen by Gaia in distances larger than about 5000 light years.

Addendum: Writing about magnitudes is always a pain in the ***, which is because the definition is kind of backwards. The magnitude of a star stands for its brightness (either apparent or absolute). So we intuitively think that a high brightness (or luminosity) also means a high magnitude. However, the magnitude system is defined with a negative sign. A bright star has a smaller numerical value for its magnitude than a fainter star. This is confusing and sometimes leads to confusing (or even plain wrong) statements. I hope I manage to avoid this in my texts.

The Milky Way: A pre-Gaia map of our home galaxy

Today's post will be about the galaxy we live in: the Milky Way. It will not be about exoplanets. However, I will come back to this topic in one of my next blogs because I will try to show where the exoplanets we know are located in this 'map' of the Milky Way.

First of all: If somebody shows you a map of the Milky Way, you should immediately be aware of the fact that this cannot be a real map in the sense that everything that is shown represents a real object with a measured position. There is no real map of the galaxy we live in, simply because (a) we cannot travel out and make a picture from above or below, and (b) we can only see 'far' enough to observe a tiny fraction of the stars in the Milky Way. The latter will hopefully improve soon because Gaia is already operating and observes more stars of the Milky Way than was ever possible before. With a little bit of luck we will get a much better impression of how our galaxy looks like this year.

What you see in the picture is my own little artist impression of how the Milky Way might look like. Again I emphasize that this might be completely wrong. Every other picture, even the I guess most famous one by NASA (R. Hurt), probably is pretty much wrong too. This 'map' is just an illustration which is supposed to show four things we believe to know about our galaxy: (a) It is a (flat) spiral galaxy. (b) It has a bright center with a bar-like structure. (c) It has four spiral arms at roughly about these locations. (d) It has a diameter of roughly about 100000 light years (ly).

Additionally, I tried to incorporate some real astrophysical data into this map. The white circles are measurements of embedded clusters (by Camargo) using the WISE telescope. The circles in cyan present the positions of molecular clouds coming from a catalogue by Ellsworth-Bowers. The only stellar data in the map are galactic cepheids (by Berdnikov, shown in magenta), which are variable stars used for distance measurements. There certainly are other dataset that should be in there to have a more complete picture, but I think these three are good enough to get the general picture. These data points give you an idea of what is actually measured and used to conclude that the Milky Way looks like what I drew.

There is a fourth dataset in the picture which cannot be seen. This dataset is the one with the best distance measurements for stars we have, at least until the first Gaia results get published. It is the Hipparcos data. However, all the data is located close to the black cross which marks the position of our Sun in the Milky Way. Our position in the Milky Way is about 8200 parsec or roughly 27000 ly away from the galactic center.

On the left side you see a blowup of the region of the Sun, where I used the new Hipparcos catalogue (van Leeuwen, 2007) to draw the positions of more than 100000 stars (white dots). Hipparcos measured the positions of stars better than one milli-arcsecond, which means the most distant stars in this catalogue are more than 3000 ly away. Of course, most of the observed stars are much closer to the Sun; 90 % of the stars with measure distances are within a radius of 1660 ly.

Although the Hipparcos map consists of a huge number of measurements, the distances from the Sun are not nearly far enough to tell us something about the large-scale structure of the Milky Way. Gaia will hopefully be about a factor 100 better than this, which will do the trick and give us a pretty good picture about a large part of the Milky Way covering maybe even 1 % of all the stars in our galaxy. Still, Gaia will not be able to see everything; some parts will be blocked from view, and some stars are just too faint or too far away to be seen.