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PH507Q
UNIVERSITY OF KENT
SCHOOL OF PHYSICAL SCIENCES
CLASS TEST 3
2016, WEEK 24
1. The gravitational perturbation of a star's position by an unseen
companion provides a signature which can be detected through
precision astrometry. Calculate the mass of an exoplanet in earth
masses which causes a 2 solar mass star’s wobble in the plane of
the sky to describe a circle of 3 microarcseconds radius and orbital
period of 2 years. Assume the star is at a distance of 20 parsecs.
First, find the distance of the planet from the dstar. Use the formula
æ m p öæ a ö
÷÷ç ÷
Dq = çç
è M * øè d ø
Here, a is the orbital semi-major axis of the planet. This has to be
calculated from the period and the star’s mass using Kepler’s generalized
law. This yields the period P =2 pi/  where
W=
G(M * + m p )
a3
to yield
a / 1AU= (P /1 year)2/3 (M / 1 solar mass)1/3
= 2AU
With d = 20 parsecs,
a/d = 4.9 x 10-7
theta = 3 microarcseconds = 3 x 10-6 x (2 pi/ (360x60x60)) radians
= 1.45 x 10-11 radians
Therefore mp / M* = 1.45 x 10-11 / 4.9 x 10-7
=
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2
3 x 10-5
So mp = 6 x 10-5 solar masses = 20 earth masses.
2. State the statistical weights for the ground and first excited
states of neutral hydrogen. For a gas of neutral hydrogen,
determine the temperature for which the number of atoms in
the first excited state is10% of the number in the ground state.
2, 8, (2n2)
1eV = 11,594K
Second level is 10.19eV above the ground. So:
Use Boltzmann equation:
exp (-10.19 x 11,594/T) = 2xN2/ (8xN1) = 0.025
yields
T = 32,000K
3. Taking Deneb to have a surface temperature of 8,500K and a radius
of 200 solar radii, determine the present spectral class and luminosity
class of Deneb. Investigate and state the stages through which Deneb is
expected to progress. State the name of the pre-main-sequence track
that Deneb would have followed.
A, Ia
20 Msun, so supergiant, core-collapse supernova
Depending on their initial masses and the rate of mass loss, they may
explode as yellow hypergiants or luminous blue variables, or they may
become Wolf-Rayet stars before exploding in a core collapse supernova.
Identifying whether Deneb is currently evolving towards a red supergiant
or is currently evolving bluewards again would place valuable constraints
on the classes of stars that explode as red supergiants and those that
explode as hotter stars.
3
As a massive protostar: Henyey or radiative track.
4. The temperature of an opaque cloud is measured to be 20K. Find the
wavelength of the peak blackbody emission. In which part of the
electromagnetic spectrum does the peak lie?
 max = 2.898 x 10-3 / T
where max is in metres when T is in Kelvin. Or:
 max 
2897
T
wheremax is in units of 10-6 m, and T is in Kelvin.
So 20K, yields a peak at 2900/20 microns = 145 microns
In the far infrared or sub-millimetre is just acceptable.
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