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The Formation and Evolution of
Planets in Protoplanetary Discs
Tutorial #1
Tilman Birnstiel, Barbara Ercolano, Giovanni Rosotti
Lesson: 18/04
Tutorial: 20/04
To be handed in before: 27/04
1. Direct imaging
Suppose you want to directly image a planet. What is the best band to
use to do such an observation?
• In the lecture you encountered the formula giving the contrast ratio
considering the light of the star scattered by the planet. Derive the
analogous formula for the contrast at the peak of the planet thermal
emission. Assume that both the spectra of the star and of the planet
can be modelled as a blackbody spectrum, with temperatures given
respectively by T∗ and Tplanet . To what value this formula evaluates
for the Earth-Sun system (assume for the Earth a temperature of 290
K)? Remember that the Plank spectrum is given by:
Bν (T ) =
1
2hν 3
,
2
c exp(hν/kB T ) − 1
(1)
where h is the Planck’s constant, c the speed of light and kB the
Boltzmann constant, and its peak falls at
hνmax = 2.8kB T.
(2)
• Unfortunately, not only the contrast makes these observations challenging, but also the spatial resolution. Consider the Earth-Sun system and work out the angular separation at distances of 5 and 50
pc. What are the sizes of the telescopes you need at the peak of Sun
emission (∼ 550 nm) or at the peak of Earth emission to resolve such
a system?
2. Astrometry
The characterist angular scale relevant for detecting an exoplanet is given
by:
Mp a
,
(3)
θ=
M∗ d
where Mp is the mass of the planet, M∗ the mass of the star, a the semimajor axis of the orbit and d the distance from the Earth.
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• Derive the previous relation.
• You can notice that the angular distance is proportional to semimajor axis of the planet. Does this always imply that planets far
from the host star are easier to detect? What other limit is there in
real observations?
• Consider the Jupiter-Sun system. Assuming an astrometric precision
of 100 µas, what is the maximum distance at which you could detect
Jupiter?
• In a real mission, the astrometric precision depends on the magnitude of the star. The future mission GAIA will have an astrometric
precision of 7 µas, but only for stars brighter than 10 mag. What
would be the maximum distance at which you could detect Jupiter
with that precision? What would be the magnitude of the Sun at
that distance? Remember: the absolute magnitude of the Sun is
4.83.
3. Habitable zone
In the lecture the circum-stellar habitable zone was defined as ”the distance from the host star where Eath-like planets would have liquid water
on surface”. You can imagine how much effort is being spent in the attempt of finding planets in this zone (why?).
• The possibility of having liquid water depends on the temperature of
the planet. Show that with very simple assumptions the equilibrium
temperature is given by the following expression:
1/4
S(1 − A)
Teq =
,
(4)
fσ
where S is the solar flux (1360 W m−2 at the Earth’s distance),
A is the albedo of the planet, σ is the Stefan-Boltzmann constant
(5.67 × 10−8 W m−2 K−4 , in SI units) and f is a constant of order
unity (assume f = 4 for the rest of the exercise, why?).
• For an Earth albedo of 0.29, derive the habitable zone around the
Sun assuming that the equilibrium temperature must be smaller than
373 K and bigger than 273 K (is it fully general to take these temperatures?). What do you note about the values you obtain?
• Derive the equilibrium temperatures for the Earth, Venus and Mars.
Use the following table for the albedos and the distances:
Earth
Venus
Mars
Distance
Albedo
Tsurface
1 AU
0.72 AU
1.52
0.29
0.9
0.25
288 K
737
218
The table also reports the actual mean surface temperatures. What
can you comment?
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Figure 1: Habitable zone in our Solar System (from Selsis et al. [2007].
• For a detailed model of the habitable zone, taking into account the
planet’s atmosphere, see Selsis et al. [2007]. Figure 1 show results for
our Solar System.
References
F. Selsis, J. F. Kasting, B. Levrard, J. Paillet, I. Ribas, and X. Delfosse. Habitable planets around the star Gliese 581? Astronomy and Astrophysics, 476:
1373–1387, December 2007. doi: 10.1051/0004-6361:20078091.
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