SPACE STATION QUIZ
1. The space station orbits Earth at an altitude of
(a) 400 km
(b)
4000 km
(c) 20,000 km
2. The time it takes the space station to complete one orbit around Earth is
(a) 1 ½ hours
(b) 5 hours
(c) 10 ½ hours
3. The speed of the space station is
(a) 0.5 km/s
(b) 7 km/s
(c) 55 km/s
4. The speed of Earth at its equator is
(a) 0.5 km/s
(b) 7 km/s
(c) 55 km/s
5. The percentage of Earth’s surface visible from the space station, at any one time, is
(a) 3%
(b)
15%
(c)
25%
6. The force of gravity inside the space station is
(a) zero
(b) .1g
(c)
.9g
Answers on next page.
1
The correct answers are: 1(a),2(a),3(b),4(a),5(a),6(c).
The answer to 1 is 400 km. Knowing this, the Earth’s radius ( RE = 6380km ) and the
acceleration of gravity at the Earth’s surface ( g = 9.8m/s 2 ), it is possible to figure out all
the other answers. Here’s how.
4. Let’s do this first because it is the easiest. A point on the Earth’s equator travels a
distance of d = 2π RE in a time of t = 24hr = (24)(3600)s . Its speed is therefore
v=
2π ( 6380 )
d 2π RE
=
=
= .5km/s , roughly
t
t
( 24 )( 3600 )
3. Now let’s figure out the speed of the space station. Since 400km is very small
compared to the radius of the Earth, let’s be rough and assume that the space station is in
a grazing orbit around Earth i.e. an orbit whose radius equals the Earth’s radius (we
assume that there are no buildings in the way, and no air resistance!). So the Space
Station orbits Earth at the uniform speed VS in an orbit of radius RE . We now use the fact
that an object in uniform circular motion at the speed v in a circular orbit of radius r has
a centripetal acceleration a given by a = v 2 / r . But the centripetal acceleration of the
Space Station if it is in a grazing orbit is just g = 9.8m/s 2 , the acceleration of gravity at
the Earth’s surface. Putting a = g , v = VS and r = RE in the previous formula gives
g = VS2 / RE , from which we can solve for the speed of the Space Station as
VS = gRE = 7.9km/s. This would be the speed if it were in a grazing orbit. At an altitude
of 400km the speed drops a little to about 7km/s. (You can figure this out from Kepler’s
3rd law, see ADDED PROBLEM on next page).
2. Let’s continue to assume that the Space Station is in grazing orbit. The time it takes to
complete one orbit is then the circumference of the orbit, 2π RE , divided by the speed
VS found in problem 3, and this works out to about 84min.
6. The gravitational acceleration produced by any body decreases as the square of the
distance from the center of that body. Let g be the gravitational acceleration at the
Earth’s surface and g h the acceleration at height h above the Earth. Then
g h / g = RE2 / ( RE + h ) , and putting in g = 9.8m/s 2 , RE = 6380km and h = 400km gives
2
g h = 8.68m/s 2 , which is about .9g.
5. It can be shown from some trigonometry and calculus that if the Space Station is a
height h above the Earth, the fraction of the Earth’s surface seen from it is h / 2 ( R + h ) ,
which works out to .029 or about 3%.
2
ADDED PROBLEM
The answers to problems 2 and 3 on the previous page were worked out by assuming that
the Space Station was in a grazing orbit around Earth. However, as I indicated in class,
one can use Kepler’s 3rd law to work out the exact values of these quantities at the actual
altitude of the Space Station. I do this here.
Let Tg be the period of the grazing orbit at the orbital radius RE . We found that
Tg = 84min . Let Th be the orbital period of the satellite when it is at height h above the
Earth’s surface. From Kepler’s 3rd Law we have that
Th2 ( RE + h )
=
Tg2
RE3
3

h 
or Th = Tg 1 +

 RE 
3/ 2
 3 h 
≈ Tg 1 +
,
 2 RE 
where the last expression was obtained by making a Taylor expansion, since h is small
compared to RE . Putting in the numbers in the last expression shows that Th = 92min.
The orbital speed at this altitude can then be calculated as 2π ( RE + h ) / Th and this works
out to 7.7km/s.
Hohman Transfer Orbit
We learnt how a spacecraft can be sent from Earth to Mars (whose orbital radius is
1.52AU) using a Hohman transfer orbit. There are three important parameters connected
with such an orbit:
(a) Transfer time: the time it takes for the spacecraft to go from Earth to the planet
(this is about 259 days for Mars).
(b) Opportunity: the angle by which the planet has to lead Earth in order that the
spacecraft successfully intercepts it (about 44˚ for Mars).
(c) Synodic Period: the frequency with which the right Earth-planet alignment for a
successful launch occurs (about 778 days for Mars).
Can you calculate these quantities for a trip to Mars using Kepler’s 3rd law and the ideas
described in class? (Solutions on the next page.)
3
Hohmann transfer orbits
(a) Let us obtain an expression for the transfer time from Earth to an outer planet. Let
RE and RP be the orbital radii of Earth and the other planet, both orbits assumed to be
circular. The spacecraft goes on an elliptical transfer orbit from Earth to the planet, the
focus of the ellipse being the sun and with the perihelion and aphelion points of the orbit
1
being Earth and the other planet. The semi-major axis of the transfer orbit is ( RE + RP ) .
2
Let TH be the transfer time in the Hohmann orbit (this is just half the period of the entire
eliiptical transfer orbit). From Kepler’s third law, the ratio of T 2 to R 3 for Earth and the
transfer orbit should be the same, giving
( 2TH )
(T )
= E3
3
{( RE + RP ) / 2} RE
2
2
⇒
1  1  R 
TH = TE  1 + P  
2  2  RE  
3/ 2
,
where TE = 365.25d is Earth’s orbital period. For Mars, RP = 1.52AU and we find that
TH = 258d .
(b) Now we calculate the angle of opportunity for a Hohmann transfer to an outer planet,
i.e., the angle by which the outer planet must lead Earth in order that the spacecraft has a
rendezvous with the planet at the conclusion of its transfer orbit. In the time TH that the
spacecraft spends in its transfer orbit, the angular distance (in degrees) moved by the
target planet is 360 / TP TH . This angle falls short of 180 because the planet moves
(
)
more slowly than the spacecraft and completes less than half of its orbit in the time that
the spacecraft traverses half the elliptical transfer orbit. Thus θ = 180 − 360 / TP TH is
(
)
the angle of “opportunity” by which the target planet should lead Earth for a successful
transfer. For Mars, TP = 684d and TH = 258d and we find that θ = 44 .
(c) Finally we calculate how often an opportunity arises. In the interval between two
opportunities, the faster moving inner planet goes exactly once more around the sun than
the outer planet. In other words, if T is the interval between opportunities, we have that
 360 
 360 
T
−



 T = 360
 T1 
 T2 
where T1 and T2 are the periods of the inner and outer planets. We thus find that the
“synodic period” T (i.e. the interval between successive moments when the sun, earth
and the planet have identical relative alignments) is given by 1/ T = 1/ T1 − 1/ T2 (= 783d
for Mars).
4
FRONTIERS: Suggestions for further reading
Most of the topics covered in the morning talks can be found in almost any college level
text. Two examples of such texts are:
UNIVERSITY PHYSICS (12th Ed) by H.D.Young and R.A.Freedman (Addison-Wesley,
2004).
PHYSICS for Scientists and Engineers by Serway and Jewett (Thomson Brroks/Cole
2004).
A very nice treatment of Einstein’s special theory of relativity is given in
SPECIAL RELATIVITY by A.P.French (W.W.Norton and Co.)
Special relativity and the fundamental ideas of atomic, nuclear and particle physics are
covered in
CONCEPTS OF MODERN PHYSICS by A.Beiser (McGraw-Hill)
A fun book that talks about hypercubes and higher dimensions is
SURFING THROUGH HYPERSPACE by Clifford A.Pickover (Oxford U Press, 1999).
Two websites on hypercubes are:
http://www.jpbowen.com/publications/ndcubes.html
http://en.wikipedia.org/wiki/Tesseract --- nice image gallery
Here are two links on Bose-Einstein condensates (BECs):
http://www.colorado.edu/physics/2000/bec/
http://math.nist.gov/mcsd/savg/vis/bec/
--- (has beautiful images)
Here are two good sites about black holes:
http://hubblesite.org/explore_astronomy/black_holes/
http://imagine.gsfc.nasa.gov/docs/science/know_l2/black_holes.html
5
FRONTIERS ’07: Physics Program
DATE
MORNING
AFTERNOON
Mon 9
Kepler’s Laws:
Sending a spacecraft to Mars
Film: Navigating in Space
Tue 10
Special theory of relativity
Film on Einstein
Physics Lab: Motion sensors,
Force Table
Wed 11
More Special Relativity
PHYSICS LUNCHEON
Liquid N2 and Physics Demos
Fri 13
Quantum Computing and
Teleportation
The physics of Imaging
(Ultrasound, Doppler etc)
Mon 16
Bohr model of the atom
Tue 17
Hypercubes and their
applications
Wed 18
Black holes and cosmology
Thu 12
6
Physics Demonstrations
Visit to WPI’s MRI Lab
Physics Lab: Atomic
Spectroscopy
Physics Lab: Waves and
oscillations, optics
Film: Stephen Hawking’s universe