# Detecting planets via radial velocity variations

You are familiar with the basic idea behind radial velocity variations, right?

The idea is pretty simple: if a star has a planetary companion, then the star will make a small orbit around the center of mass of the star-planet system. As the star revolves around this center, its light will be Doppler-shifted with the period of its orbit. Find this periodic shift, and *boom* there's the planet.

So, what's so hard about that?

```

Q:  Really -- why is this so difficult?

```

The problem is twofold:

• Planets are (in general) much, much less massive than their host stars. This means that the center of mass is very close to the star, which means that the size of the star's orbit is small, which means that the star's motion is SLOW.
• The Doppler shift depends (to first order) on the ratio of this orbital speed to the speed of light; and the speed of light is FAST.

Let's do a simple example: imagine a simpler Solar System which consists of just the Sun and one planet: either the Earth, or Jupiter.

```
Earth           Jupiter          Sun
---------------------------------------------------------------------
Mass (kg)               5.98E24           1.90E27        1.99E30

Orbital radius (m)      1.496E11          7.78E11

---------------------------------------------------------------------

Q:   What would be the speed of the Sun's motion around the
center of mass in each case?

```

Okay, so the speed of a star's orbital motion around the center of mass isn't going to be very large. What does that mean for the Doppler shift?

```

Q:   You observe one absorption line in a stellar
spectrum, which has a rest wavelength of 5000 Angstroms.

If this star is moving away from us at 10 m/s due to the
gravitational pull of a planet, what will
the observed wavelength of this line be?

```

That's a pretty small shift. Here, for example, is a typical spectrum of a G2V star.

There's no way to measure a 10 m/s shift in a spectrum like this, which is low-resolution. Recall that physicists describe a spectrograph by its resolving power:

```
central wavelength
resolving power  R   =    -------------------------------
smallest feature visible

```

In the examples above, the resolving power is only about R = 1000. That's typical of general-purpose instruments: the SDSS spectrograph had R = 2000.

People interested in planets build spectrographs with higher resolving powers.

Compare a spectrum from HARPS to one from a "typical" spectrograph; these are two different stars, but you should get the general idea.

But even these special-purpose instruments don't have nearly the resolving power to detect the shift in a spectral line caused by a star's reflex motion in response to a planet ... do they?

Actually, they do ... if you use them properly. There are two "tricks" in the analysis of stellar spectra which allow astronomers to measure Doppler shifts of just a few meters per second.

```

```
A. Measuring the centroid of a line to a sub-element level
Even though the spectrum may be measured in chunks of size L Angstroms, if it is properly sampled, one can determine the center of each line with an uncertainty which is less than L.

The general idea of determining the centroid of a sampled quantity to a precision higher than that of the samples themselves is the same reason that people can use ground-based optical images to measure the positions of stars to better than 1 arcsecond.

B. Measuring LOTS of lines
Look again at that tiny little section of the spectrum: there are at least 4 good lines in a section of just about 0.4 Angstroms.

```
Q:  How many good lines might appear in the entire
HARPS spectrum?

```

In their paper describing the discovery of the planet around 51 Peg in 1995, Mayor and Queloz write:

Let's put these two "tricks" together and see if we can figure out just how well a spectrograph like HARPS ought to do:

```

Range from 4000 - 7000 Angstroms, sampled every 0.01 Angstrom.
Use as the 'typical' wavelength 5000 Angstroms.

- determine centroid of each line to 0.2 elements

- 3 good lines per Angstrom, on average

Q:  What is the uncertainty in wavelength for one good line?

Q:  What is the uncertainty in wavelength for ALL good lines,
combined together?

Q:  What is the uncertainty in radial velocity, using all
the lines combined together?

```

#### Echelle spectrographs

Great! It appears that if we can take a spectrum which records tens of thousands of absorption lines, all sampled very finely, then we can determine the radial velocity of a star to the required precision.

Wait a moment ... if there are tens of thousands of lines, and each one is sampled finely, and there's lots of space between each one, then .... um, well, the spectrum must contain hundreds of thousands of elements.

But -- how can they measure it? Don't even the largest CCDs have only 8192 pixels in each row? If we try to sample the entire spectrum

Image courtesy of David Haworth's "Observational Astronomy" site

onto a single CCD, only a teeny tiny portion will fit!

```

Q:   How can we record the entire spectrum in one exposure?

```

The answer is to use not one dispersing element (prism or grating), but two dispersing elements oriented at right angles. One spreads the light out left-right,

the other spreads the light out up-down,

so if you use both in sequence, you get many little rows running left-right, down a bit, left-right, down a bit, and so forth.

In other words, an echelle spectrograph produces a rectangle of spectrum, with many horizontal rows inside. This rectangle will fit very neatly inside a typical CCD, so we can get a very large stretch of spectrum with a high resolution in a single exposure.

Solar spectrum courtesy of Kitt Peak National Observatory and the Harvard-Smithsonian Center for Astrophysics

Well, actually, the picture above is not an actual single exposure -- it's a processed and rectified pretty picture. An ACTUAL single echelle image looks more like this:

Spectrum of 51 Peg taken by the ELODIE instrument

or this (which is actually an image taken by amateurs with a commercially available echelle spectrograph )

Image courtesy of Olivier Thizy and ASPA

As you might imagine, reducing and analyzing the data recorded in echelle images is ... not so easy. There are a LOT of non-linear effects that must be calibrated and removed.

Some of those effects occur due to changes in the instrument as is expands or contracts or bends. Obviously, astronomers want to prevent such instrumental changes from happening. In the case of exoplanet searchers, this quest for stability has led to

• the spectrographs leaving the back of the telescope,

Image of Brashear spectrograph at Lick Observatory courtesy of the Smithsonian

and instead sitting motionless in a room, fed by long fibers from the telescope.

Image of HARPS courtesy of ESO

• spectrographs being housed in temperature-controlled environments: the room for HARPS keeps the temperature variations to less than 0.1 Celsius.

Image of HARPS courtesy of ESO

• clever means for calibrating the light, such as a cell of gas with many absorption lines placed in the optical path. An early choice was hydrogen fluoride

Text from Campbell and Walker, PASP 91, 540 (1979)

but later, iodine was chosen as the standard.

Image of HARPS courtesy of ESO

Image from a presentation by Valenti and Anderson (a copy available below)

Image from a presentation by Valenti and Anderson (a copy available below)