Download The Sun and Planets Homework Solutions 4.

The Sun and Planets
Homework Solutions 4.
Spring Semester 2017
Prof Dr Ravit Helled
Due by: 23.03.2017
TURN IN YOUR SOLUTIONS NEXT WEEK IN CLASS
or
EMAIL THEM AS A PDF
Exercise 1.
Keplerian Orbits
Calculate the following quantities for the orbits below: periastron and apoastron distances,
minimum and maximum orbital speeds, and orbital period. Report your distances in AU,
speeds in km/s, and periods in days.
a) Earth orbits the Sun at a cozy average distance of 1 AU with a relatively small
eccentricity of 0.0167.
b) PSR J1719-1438 b holds the record for the smallest orbit at 0.004 AU. It orbits
millisecond pulsar1 and is most likely made of crystalline carbon, a material with
far greater density than diamond. The eccentricity of its orbit is not known precisely,
we only know that it is less than 0.06. For this problem, assume a semi-major axis
of 0.004 AU and an eccentricity of 0.06.
c) HD 20782 b is on a nearly hyperbolic orbit. It has the largest known eccentricity of
any planet at 0.97 ± 0.01. It orbits a solar-like at a distance of 1.36 AU.
d) A hypothetical planet orbits a hypothetical star at a hypothetical distance of 14 AU
with a hypothetical eccentricity of 0.999.
1
A pulsar is a highly magnetized, rotating neutron star or white dwarf. They are the remnants of a
supernova explosion.
1
Exercise 2.
The Greenhouse Effect
By now you already know that Venus is kept warm by a very strong greenhouse effect.
Let’s find out just how strong. Effective temperature is roughly the average temperature
that a planet’s surface would be in the absence of an atmosphere. The following formula
can be used to calculate effective temperature Te :
LSun
(1 − A) = 4σTe4
2
4πa
(1)
In this equation, LSun is the Sun’s luminosity, which is equal to 3.85 × 1026 W (watts), a
is the planet’s semi-major axis, A is called the planet’s albedo and is a measure of how
reflective the planet’s surface is (a black surface would have an albedo close to zero while
a reflective mirror’s albedo is 1), and σ is a physical constant called the Stefan-Boltzmann
constant, which has a value of 5.67 × 108 W·m−2 K−4 .
a) The surface of Venus has albedo of 0.75. What would the surface temperature be
if the planet had no atmosphere? How many degrees hotter is Venus in reality, due
to the greenhouse effect?
b) The surface of Earth has albedo of 0.28. How many degrees hotter is Earth, due to
the greenhouse effect, than it would be without an atmosphere? Which planet has
the stronger greenhouse effect, Earth or Venus?
c) There is about as much CO2 on Earth as there is on Venus. Why don’t the two
planets have similar greenhouse effects?
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Answers:
1. The equations needed to solve parts (a) through (d) are as follows:
rperi = a (1 − e)
rapo = a (1 + e)
s
v=
GM?
s
P =
2 1
−
r a
4π 2 a3
G (M + m)
a)
rperi
rapo
vmin
vmax
P
= 0.9833 AU
= 1.0170 AU
= 29.28 km/s
= 30.29 km/s
= 365 days
rperi
rapo
vmin
vmax
P
= 0.00376 AU
= 0.00424 AU
= 543 km/s
= 612 km/s
= 0.0754 days
rperi
rapo
vmin
vmax
P
= 0.0408 AU
= 2.6790 AU
= 3.2 km/s
= 207 km/s
= 579 days
rperi
rapo
vmin
vmax
P
= 0.014 AU
= 27.99 AU
= 0.26 km/s
= 616 km/s
= 110 050 days
b)
c)
d)
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2.
a) The equation for calculating the effective surface temperature in the absence of
an atmosphere was provided. All you needed to do was look up the semi-major
axis of Venus (a = 0.723). The resulting effective surface temperature is
Te = −41.44◦ C (231.71 K)
The actual surface temperature on Venus is Tactual = 462◦ C. The greenhouse
warming is found by taking the difference between the actual temperature and
the value calculated from the provided equation. The result is a greenhouse
effect of more than 500◦ C.
Tactual − Te = 462◦ C − (−41.44◦ C) = 503.44◦ C
b) The semi-major axis of the Earth is of course 1 AU. The resulting effective
surface temperature is
Te = −16.43◦ C (256.72 K)
The actual mean surface temperature on the Earth is only around Tactual =
15◦ C. Therefore the greenhouse effect is relatively weak,
Tgreenhouse = 15◦ C − (−16.43◦ C) = 31.43◦ C
c) The CO2 on Earth is locked in the oceans and in carbonaceous rocks, whereas
on Venus the CO2 is mostly in the atmosphere.
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