SOLUTION SHEET – 2007
UNIVERSITY OF KENT
SCHOOL OF PHYSICAL SCIENCES
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PH709
L7
MDS
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(a) Explain the difference between the transit technique and the microlensing technique,
describing the nature of the change in flux and the physical information which is
directly derivable from each.
Transits result in a decrease in flux of the parent star through direct blocking.
They are repeated events.
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Change in flux F is described by a light curve of the form:
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The depth of transit = fraction of stellar light blocked. In terms of the planet and stellar radii:
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Derivable information:
Hence this is a measure of planetary radius.
Duration of transit gives measure of the orbital radius.
We also obtain information on the inclination (ingress and egress provides accurate value)
Bottom of light curve is not actually flat, providing a measure of stellar limb-darkening
Deviations from profile expected from a perfectly opaque disc could provide evidence for
satellites, rings.
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SOLUTION SHEET – 2007
UNIVERSITY OF KENT
SCHOOL OF PHYSICAL SCIENCES
Microlensing planets leads to a sharp peak in flux as a background star is ‘lensed’ by the star
and planet, the light being bent according to General Relativity.
The event is one-off and planet discovery requires confirmation by other techniques.
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The flux changes as follows:
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Derivable information:
The planet mass can be loosely constrained: The relevant time scale is called the Einstein time
and it's given by the time it takes the lens (moving at speed v) to traverse an Einstein radius.
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SOLUTION SHEET – 2007
UNIVERSITY OF KENT
SCHOOL OF PHYSICAL SCIENCES
(b) A planet of radius 0.05 Rsun orbits a main-sequence K star of radius 0.8 Rsun and mass 0.8
Msun at an orbital radius of a = 0.5 AU. Estimate the probability of a transit being observed for
such a system if the inclination, i, of the planet’s orbit to the plane of the sky is distributed
randomly over 4steradians
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(full angle is 2 theta)
For a planet with radius rp << R*, probability of transits is:
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ORProbability = 2 pi x 2 / 4 
Either way yields a probability of 0.8 Rsun/0.5 AU
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= 0.0075
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SOLUTION SHEET – 2007
UNIVERSITY OF KENT
SCHOOL OF PHYSICAL SCIENCES
=
(c) If the inclination i = 90°, calculate the full duration and the depth of an observed transit.
The orbital velocity of the planet can be derived from the centrifugal force balance
with gravity
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This yields V2 = GM*/a
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= 37.9 km/s
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Therefore the full duration of the transit event will be (4Rp + 2R*)/V
Not forgetting the ingress and egress.
=
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2
33,250 seconds or 9.2 hours
The depth of the transit is
100 (Rp/R*)2 %
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= 100 x (0.05/0.8)2 = 0.39%
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