ORBITAL MOTION

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ORBITAL MOTION
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Formulas
Aphelion, perihelion refer to farthest distance,
and closest to the Sun. Apogee, perigee refer to
farthest distance and closest to the Earth. (peri
is the closest)
Finding the semi major axis. This is the average
distance from the orbiting body
How much it departs from a circle between o and 1
Velocity at peri (closest point), use correct
Velocity at ap (farthest point), use correct
Period using a in solar system only use if
orbiting the Sun.
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Period of orbiting object in seconds, use correct
The escape velocity from a body , use correct
Velocity for object orbiting in a circular orbit
. Use correct
Closet distance the orbiting object comes to the
object being orbited in an elliptical orbit
Farthest distance the orbiting object comes to
the object being orbited in an elliptical orbit
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Conversions
To change from km/sec to miles/hr Km/sec (3600
sec/hr)(0.62137 miles/km miles/hr
If the period is in seconds / 3600 for hours
To change from au to km multiply by 150,000,000
or km
To change from miles to km multiply by 1.61
km/mile
e has no units.
There is a review of math with scientific
notation on slide 15
Be sure you use the of the body that is
being orbited
I did not always follow exactly significant
figures, but I did not use all the digits.
Answers may vary slightly depending upon how you
round off the decimals, and thats ok.
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The gravitational parameter
In Astrodynamics,the standard gravitational
parameter    of a celestial body is the product
of the gravitational constant     and the mass
                 The units of the standard
gravitational parameter are km3 s-2

Be sure to use the right for the object
you are orbiting !
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Eccentricity of ellipse

• eccentricity (e) 0circle, 1 line
• aphelion - perihelion

___________________________
e
aphelion perihelion
e0
e0.98
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Properties of Ellipses
semi-major axis a 1/2 length of major axis
b
a
a Semi-major axis b Semi-minor axis
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a aphelion perihelion
_______________________
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aphelion
perihelion
P2 a3
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Using the KeplersThird Law

• P2 µ a3
• P2 a3 if
• P measured in earth years, and a in AU.
• A planets avg distance from the sun is 4 au,
  what is the period of the planet ?

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Astronomers use the metric system, whereas we are
all more familiar with the English system. I will
use conversion so that you can be more familiar
with the answers. I. The space shuttle is in a
circular orbit 200 miles above the earth. Find
the period and velocity of the shuttle. 200miles(1
.61 km/mile) 322 km above the Earth. The height
of the satellite in the problem must be the
radius of the earth height of object. Radius of
earth 6378km rh 6378322 6700 km
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You cant use because it
circles the earth, not the Sun.
a is the distance from center of Earth to the
shuttle
II. This problem covers a lot of formulas. An
asteroids closest approach to the sun is 2 au,
and its farthest distance from the Sun is 4.5 au.
Find a, the eccentricity, distance at perihelion,
distance at aphelion, period, velocity at
perihelion, and aphelion.
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Find the perihelion, and aphelion distances.
3.25au (1 - 0.385) (3.25)(.615) 1.99 au
3.25au (1 0.385) 4.43 au
Find the period.
Perihelion, and aphelion must be changer to km,
since contains km . To change multiply au
by 150,000,000 km/au , or
24.8 km/sec(3600)(.6213) 55,469.7 miles/hour
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3.504(3600)(0.62137) 7,838.8 mils/hour
III. An Earth satellite is in an elliptical orbit
around the Earth its perigee is 160 km, and the
apogee is 800 km. Find e, period, and velocity at
perigee, and apogee. Radius of the Earth 6380 km
800 6380 7180km
160 6380 6540 km
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We need a
/3600 1.57 hrs
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I had to include my Halleys comet problem.
Halleys has a period of 76 years and e0.967
Find the velocity at perihelion (close), and
aphelion (far).
17.94(0.033) .592 au ( )
17.94(1.967) 35.45 au ( )
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A Review of Scientific Notation Math

• Multiplication
• The digit terms are multiplied in the normal way
  and the exponents are added. The end result is
  changed so that there is only one nonzero digit
  to the left of the decimal.
• Example (3.4 x 106)(4.2 x 103) (3.4)(4.2) x
  10(63) 14.28 x 109 1.4 x 1010(to 2
  significant figures)
• Example (6.73 x 10-5)(2.91 x 102) (6.73)(2.91)
  x 10(-52) 19.58 x 10-3 1.96 x 10-2(to 3
  significant figures)

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• Division
• The digit terms are divided in the normal way and
  the exponents are subtracted. The quotient is
  changed (if necessary) so that there is only one
  nonzero digit to the left of the decimal.
• Example (6.4 x 106)/(8.9 x 102) (6.4)/(8.9) x
  10(6-2) 0.719 x 104 7.2 x 103(to 2
  significant figures)
• Example (3.2 x 103)/(5.7 x 10-2) (3.2)/(5.7) x
  103-(-2) 0.561 x 105 5.6 x 104(to 2
  significant figures)

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• Powers of Exponentials
• The digit term is raised to the indicated power
  and the exponent is multiplied by the number that
  indicates the power.
• Example (2.4 x 104)3 (2.4)3 x 10(4x3) 13.824
  x 1012 1.4 x 1013(to 2 significant figures)
• Example (6.53 x 10-3)2 (6.53)2 x 10(-3)x2
  42.64 x 10-6 4.26 x 10-5(to 3 significant
  figures)
• Roots of Exponentials
• Change the exponent if necessary so that the
  number is divisible by the root. Remember that
  taking the square root is the same as raising the
  number to the one-half power.
• Example