Radial velocity
physics:
In 1687, Newton stated in his Principia, that masses attract
each other with a force, also known as the law of universal gravitation:
(1)
F = G*M*m / r*r
G = Gravitational Constant
r = distance between masses m and M
A second law of physics, applicable on a planetary system is Conservation of Momentum:
(2)
m * v (planet) = M * V (star)
v, V = absolute velocity planet, absolute Velocity star
m, M = mass planet, Mass star
Now which masses m of planets can we detect?
Then we have to know the three other quantities (v,V and M) in (2).
This can be done:
(5) m * sin ( i ) = M * Vrad/v
On the right side we see the observables. So what can be deduced is
the left side, not m, but m*sin(i). Now our previous
question can be answered. Current techniques can achieve a precision
in Vrad of 3m/s. If we assume M=Msun, sin (i)=1, vmin = 10m/s and P=1 year
(so simular to earth) then:
m = 0.0003*M = 0.3* Mjupiter.
With the radial velocity method there have been found 8 candidates.
Some of these candidates have a very small r and high m, totally unlike
our own planetary system. One of them: Peg 51 was observed with higher
precision by Gray . He concluded that the Doppler shift was merely a change
of shape of the spectral lines rather then a Doppler shift. (Gray, 1997).
However, results of observations by other groups (see new
results ) were negative.
On Mark's page we can see a
table of already found planets. For these planets M is approximately 1*Msun.
There are roughly three groups of planets: the four 'Gray' planets at 0.2
AU, two planets at 0.5 AU and another two at 2.0 AU.
The equation (5) tells us that from the measurements only a minimum
mass for the planet can be inferred, so all candidates could have higher
masses. If a different method can detect this planet also, maybe we can
be sure about the mass. This brings us to astrometry.
Astrometry
By carefully looking at the position of stars, called astrometry, it
may be possible to see a change in position of a star, with a candidate
planet. Astrometry has been done for thousands of years, since the Babylonians
made their first star maps. Tycho Brahe (1546-1601) achieved a precision
of 0.5 arcmin. In twentieth century, the dutch astronomer P. vd. Kamp,
was the first to search for planets surrounding other stars, using astrometry.
He claimed two planets surrounding 'Barnards' star, but these observations
have not been confirmed. Possibly it was an instrumental error, since the
errors of the measurements (0.006 arcsec) were nearly as large as the measurements
themselves (0.02 arcsec).
physics:
Observations again give M. P is determined from the periodically change
in position: a nearby single star moves in a straight line along the sphere
of the heavens. Approximately a few arc-seconds a year. If the star has
a companion then the projection of the movement looks like a moving snake:
~~~ and this gives us the period P. So we can use equation (3) again to
calculate r. Now substitute r in (4) to obtain a value for v.
Because inclination does not matter for this method we can rewrite
(5): m*v = M*V.
After substituting v (=2*pi*r/P) and V (=2*pi*R/P) in (5) and dividing
by 2*pi/P this equation becomes: m*r = M*R, with R the change in position
of the star, caused by the planet.
If D is the distance of the star, the change in position is given by the angle: theta = 2*R/D. So now we have:
(6) m = (0.5*theta*D*M) / r
On the right side we see the determinated quantities. In the derivation of this equation it was again assumed that the planet has a circular orbit. A reasonable accuracy (in modern times) is 0.002 arcsec. So if we look at a sunlike star, distance 5pc, with a planet at 1AU then mmin = 5*mjup.
future:
There are plans for a astrometry-space-mission called Gaia.
This sattelite should reach a precision of 0.00001 arcsec = 10 microarcsec
and should certainly find planets.
So astrometry is a way to determine exact masses of planets. Another
indirect way is :
Occultation method
When a planet passes in front of its parent star it blocks a small
fraction of the light from that star. To obtain a period the star must
be watched for a second transit.
However the observer must be aligned with the planet-star system, and
the chance this happens is < 1%. If we monitor 5000 stars for
4 years than we should find 25-50 planets. The third and subsequent transits
all with the same period, duration and change in brightness determine the
planet-distance to the star and the size of the planet (from the fraction
of the diminished starlight the projected area of the planet is calculated).
If the method is sensitive enough; the technique should detect the diminishing
of the stellar brightness by 1 part in 10000, earth mass planets could
be detected. The Kepler
mission uses this technique and the sattelite is theoretically capable
to detect Earth-sized planets.
According to Einstein' s theory of relativity, mass bends light. The result of this: if a star (lensing star) passes before a more distant star (source star), a double image is formed, and the image is magnified and amplificated. This effect is called the lensing effect.Maximum amplification occurs when the source star is right behind the lensing star. Then a ring is formed, instead of a double image. The diameter of this ring is
(7) ((4*G*M/c*c) * Dds*Dd/Ds)^1/2 , where:
c = speed of light.
Dd= distance to lens
Ds = distance to source
Dds = Ds - Dd
Because the diameter is usually in the order of micro-arcsec we speak
of microlensing. Microlensing gives only two quantities which can be measured:
the amplification and the duration of this amplification. A typical microlensing
event has a duration of 40 days.
The chance of seeing the effect (called optical depth) is very small,
so we have to look at many stars to see it. A good idea is to look at the
core of the Milky Way (in this case we also know Ds = 8 kpc, since all
core-stars are located at the same distance, see Stellar
populations), where the optical depth is 1 out of 1000000. So if we
look at 1000000 stars directly, we expect to see one event. In the past
five years, technology developed and we are now able to look at these huge
amounts of stars simultaneously. About 100 times have been reported at
this moment.
If the lensing star has a companion we see two peaks in brightness. When this companion is a planet, the timescale of the planet-lensing effect is in the order of days for Jovian-planets and hours for terrestial planets. More precisely:
(8) t = 1.7 hours * ((Mplanet/Mearth)^0.5) * ((Mlstar/Msun )^-0.5)
So if we want to see a terrestial planet, then observations have to be made almost continuously. This requires telescope time and new techniques, but may become possible before 2000. Currently, several teams of astronomers are working on microlensing, resulting in the detection of binary stars. The PLANET collaboration, which has a high time-resolution is specialised in detecting planets. Within one or two years, it should be possible to detect Jovians.
The problem with direct detection is that the companion overshines the
orbiting planets. Therefore the light of the star has to be obscured. It
is best to look in IR wavelenghts because the IRplanet/IRstar ratio is
1000 x higher then the visible light ratio.
The next problem then is to have sufficient angular resolution to separate
the planet and the star. This problem is solved if we use an interferometer,
which is a combination of telescopes.
Another problem is that the light of the planet is very faint, so there
has to be corrected for background radiation, such as:
-) diffracted starlight, eleminated by cooling the telescopes.
-) solar system IR zodiacal emission, caused by the reflection
of sunlight of dust in the solar system. This can be solved if the interferometer
is placed at 4-5 AU.
A proposal for a satellite which combines all these informations is
the Darwin satellite.
This device is capable of detecting directly terrestial planets, taking
spectra of these planets and finding an atmosphere. Because life betrays
itself by characteristic properties of an atmosphere, Darwin may even answer
the thousands of years old question, whether and where there is other life
in the universe.
References: