# 7.5: Astrometry

Astrometry is one of the oldest methods that astronomers have for studying the motion of stars. The name says it all: astrometry is the process of measuring the positions of objects in the sky. This technique has been used to measure parallax (distances) to stars, proper motions (motions in the plane of the sky) and to detect binary star orbits. More recently, astronomers have tried to detect exoplanets with this technique. This technique has not been very successful from ground-based telescopes because distortions from the atmosphere ("twinkling" of stars) blurs out the spatial position of stars and therefore does not permit precise enough measurements. However, astrometry as a planet detection technique is about to explode onto the scene with the launch of the Gaia spacecraft by the European Space Agency.

The basic idea behind astrometry for exoplanet detection is depicted in this NASA animation. Astronomers obtain a series of pictures over time (a "time series") and use at least three reference stars (which have nearly constant positions) to measure the changing position of the host star in the plane of the sky.

Q1. How is astrometry (or the "astrometric technique") different from Direct Imaging or from the Doppler method?

With direct imaging, astronomers obtain an actual image of the planet. To confirm that the planet is not a background source, additional observations are taken over time, verifying orbital motion of the planet candidate. But the big difference between direct imaging and astrometry is that the planet is observed.

Like the Doppler (or "radial velocity") technique, astrometry is an "indirect" method - the planet is not seen, but it's existence is inferred by the gravitation effect that the planet exerts on the host star. However, different things are being measured with Doppler and Astrometric techniques; the radial velocity technique measures the changing speed of the star along one dimension: the line of site. The fundamental data are measurements of radial velocity over time. Astrometry measures the changing position (not speed) of the star projected onto the two dimensional plane of the sky. The fundamental data are positions (angles and angular separations) over time.

Because astrometric measurements are made in two dimensions (x, y in the plane of the sky) the astrometric orbital model measures the true mass of the planet; it does not suffer from the unresolved inclination of orbital models from the radial velocity method.

The changing position of the host star is determined by the mass and the orbit of the planet. The main equation to keep in mind is:

$M_{star}\ast\alpha_{star}\,=m_{planet}\ast\alpha_{planet}$

Where:

• $$M_{star}$$ is the mass of the star
• $$\alpha_{star}$$ is the angular separation between the star and the center of mass
• $$m_{planet}$$ is the mass of the planet
• $$\alpha_{planet}$$ is the angular separation between the (unseen) planet and the center of mass.

The first thing to do is to use the distance to the star to convert $$\alpha_{star}$$ and $$\alpha_{planet}$$ into physical distances in Astronomical Units, or AU's. Referring back to the section on measuring stellar distances, use:

$$r_{star}\,=\alpha_{star}\ast D_{star}$$

Where $$r_{star}$$ is the distance between the star and the center of mass in AU and $$D_{star}$$ is the distance from our solar system to the star in units of parsecs. Then, equation $$\PageIndex{1}$$ becomes:

$M_{star}\ast r_{star}\,=\,m_{planet}\ast r_{planet}$

Q2. Calculate the Astrometric Displacement of the Sun

1. The mass of Jupiter is 0.001 times the mass of the Sun and Jupiter orbits at a distance of about 5 AU from the Sun. What is the astrometric displacement (in AU) of the Sun because of Jupiter?
2. The mass of Saturn is 1/3 the mass of Jupiter and Saturn orbits at a distance of about 10 AU from the Sun. What is the astrometric displacement (in AU) of the Sun because of Saturn?
3. If Saturn (at 10AU) had the same mass as Jupiter, would the astrometric displacement of the Sun be smaller or larger than the displacement from the real Jupiter at 5AU?
4. Can you picture the total astrometric orbit of the Sun that is caused by both Jupiter and Saturn?

Notice that even if you know the mass of the star (from its spectral type) and the distance between the star and the center of mass with astrometry, there are still two unknowns in equation 2: the mass of the planet and the semi-major axis of the planet. However, astrometric observations also reveal the orbital period so we can use Kepler's Laws to derive the semi-major axis of the planet. Then, if we know the mass of the star and measure the astrometric displacement of the star, you can solve for the planet mass.

## Astrometry of multi-planet systems

Consider our solar system. With 8 planets, this is a good example of a multi-planet system. What does Kepler's Third Law tell you about the orbital periods of our planets? The star wobble demo in the downloaded NAAP package (Extrasolar Planets simulations) will allow you to add in an arbitrary number of solar system planets and see the gravitational effect on the Sun that might be observed if we were observing from above the orbital plane. However (this is an important point), to fully model an astrometric orbit, astronomers must observe the star through a large fraction of one complete orbit. This would require about 12 years of observations to detect Jupiter or 30 years for Saturn. How long will the Gaia mission fly?

Q3. Astrometric Displacement of the Sun - a Bird's Eye view

Run the NAAP simulator, selecting only Jupiter. Then run the simulation selecting only Saturn. Does this match the result you expected from Q2 parts a and b?

Now run the NAAP simulator selecting both Jupiter and Saturn. Does this result match your picture from Q2 part d?

How do the astrometric influences of Mercury, Venus, Earth, Mars and Pluto compare to the influence of Jupiter, Saturn, Uranus and Neptune.

The astrometric motion of the Sun is depicted in the figure below. These positional measurements assume a birds eye view looking down onto the orbital plane of the solar system.