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All planetary orbits contain 5 unusually stable points. These points are particularly important because they allow manmade satellites to orbit the Sun with a period equal to that of Earth's. 3 of these points are collinear. Suppose that is the distance between the centers of mass of Earth and the Sun.
Find the distance from Earth's center of mass to either one of the other stable points in the EarthSun system in terms of…
(I'm not looking for a full solution; I just want to know what these points are called)
What are these points it talks about, and what's their mathematical relation to Earth's orbit?
This still isn't homework; I'm just looking for the name of these points.
These are the Langrangian Points http://en.wikipedia.org/wiki/Lagrangian_point you'll find an explanation of the maths at a variety of sources if you search using that.
As Jeremy explained, these are the Lagrangian points (see his link to the Wikipedia article). At these points, earth's gravity and the sun's gravity partially cancel each other to cause the orbital period of an object at that point to match the orbital period of the earthsun system.
Satellite/Planetary Orbits  Astronomy
The Gravitational Force varies in proportion to the Mass of the attracting body and inversely with the square of the distance to the body. The Sun is much more massive than any of the planets and its gravity dominates the Solar System. Only quite near the planets, does the planetary gravity become stronger than that of the Sun.The first reaction is to determine the point between Sun and Planet where their gravitational forces cancel out. This would be OK if we were not considering a moving objects. Since there is orbit acceleration as well as a gravitational one, these have to be considered also. Laplace derived the following for the Radius of the Sphere of Influence
 D SP = the distance between the Sun and the Planet
 M P = mass of the planet
 M Sun = mass of the Sun
Example: Jupiter and Earth
The Sun is 1047 times more massive than Jupiter (M Sun /M J = 1047). The distance between the two is 5.02 AU (AU = Astronomical Units = 150 million km). So
r J = (5.2 x 1.5x10 8 km) [1/1047] 2/5 = 48.3 million km
The most distant satellites of Jupiter are about 1/2 this distance away from the planet. Jupiter's radius is 73,500 km
so in terms of the planets radius, R Jupiter : r J = 657 R Jupiter
Earth:
Earth is 1/333,000 as massive as the Sun and only 1 AU away. It's sphere of influence is thus:
r E = 1.5x10 8 km [1/333,000] 2/5 = 927,000 km
= 2.4 x distance from Earth to the Moon
= 145 R Earth (145 times the radius of the Earth)
Relative Motion
In order to properly describe the motion of the probe with in the sphere of influence of the planet, we must know its velocity with respect to the planet. The motion in the solar sysem is known relative to the Sun and we must transform the velocity relative to the Sun, v o to one relative to the planet, v. We must know the velocity of the planet, v p , relative to the sun. We must also know where the space probe enters the sphere of influence. The diagram to the right illustrates the parameters involved.
The velocity relative to the planet is given by subtracting the planet's velocity from the space probe's.
Since velocity has direction and magnitude, we need to determine both. The result of a little trigonometry gives
v x = v p  v o cos A o
v y = v o sin A o
The angle of the new velocity is
tan(A) = v y /v x
The magnitude of the velocity is then determined using the components
 v  = [v p 2 + v o 2  2 v p v o cos A o ] 1/2
Hyperbolic Orbits
Since the space probe enters the sphere of influence with a net velocity inward, the probe will have a velocity relative to the planet greater than the escape velocity of the planet. This produces a hyperbolic orbit within the sphere of influence. All orbits (elliptical, parabolic, and hyperbolic) are described geometrically by,
Where
r = distance from the center of the planet at angle f
e = eccentricity of the orbit ( e > 1 for a hyperbolic orbit)
a = semimajor axis for elliptical orbit,
distance from closest approach to the intersection of the asymptotic lines for hyperbolic orbits
f = position angle relative to perihelion
When the probe is very far from the planet (entering the sphere of influence), then it is at its maximum angle , f m .
e cos f m + 1 = 0
and this gives a relationship with the eccentricity of the orbit.
e = 1/cos f m
The impact parameter, b, is the distance from the planet that the probe would have it travelled in a straight line inside the sphere of influence. This parameter along with the velocity, v, determines the deflection angle, 2 q .
M = mass of the planet
G = Universal Gravitation constant = 6.67x10 11 N kg 2 /m 2
The maximum angle and the deflection angle are related by
f m = p /2 + q
Hence the eccentricity is e = 1/sin q
These relationships will be important in deternining the effect of the planet on the space probes orbit. One useful property of the orbits is that they are symmetric about the semimajor axis. The velocity of infall before perijov (closest point of approach to Jupiter) will have the same magnitude as the outgoing velocity at the same distance from perijov. We will use this in our example.
Для показа рекламных объявлений Etsy по интересам используются технические решения сторонних компаний.
Мы привлекаем к этому партнеров по маркетингу и рекламе (которые могут располагать собранной ими самими информацией). Отказ не означает прекращения демонстрации рекламы Etsy или изменений в алгоритмах персонализации Etsy, но может привести к тому, что реклама будет повторяться чаще и станет менее актуальной. Подробнее в нашей Политике в отношении файлов Cookie и схожих технологий.
Satellite/Planetary Orbits  Astronomy
The following tables list the current JPL planetary satellite ephemeris files and their corresponding accuracies. These ephemeris files are used by HORIZONS to provide highprecision ephemerides for the planetary satellites. Tables are grouped by planet. Table column headings are described below.
Martian Satellites
Sat.  Code  Ephem  Start Time  Stop Time  R (km)  T (km)  N (km)  P (s)  RA/Dec (arcsec)  Ref. 

Phobos  401  MAR097  01Jan1600  04Jan2500  2  5  2  0.0001  0.01  1 
Deimos  402  MAR097  01Jan1600  04Jan2500  3  10  3  0.0003  0.02  1 
Galilean Satellites of Jupiter
Sat.  Code  Ephem  Start Time  Stop Time  R (km)  T (km)  N (km)  P (s)  RA/Dec (arcsec)  Ref. 

Io  501  JUP310  18Dec1900  14Jan2200  5  5  5  0.0003  0.002  3 
Europa  502  JUP310  18Dec1900  14Jan2200  5  5  5  0.0003  0.002  3 
Ganymede  503  JUP310  18Dec1900  14Jan2200  5  5  5  0.0003  0.002  3 
Callisto  504  JUP310  18Dec1900  14Jan2200  5  5  5  0.0003  0.002  3 
Jovian Inner Satellites
Sat.  Code  Ephem  Start Time  Stop Time  R (km)  T (km)  N (km)  P (s)  RA/Dec (arcsec)  Ref. 

Amalthea  505  JUP310  18Dec1900  14Jan2200  200  400  100  0.0003  0.13  3 
Thebe  514  JUP310  18Dec1900  14Jan2200  200  400  100  0.0005  0.13  3 
Adrastea  515  JUP310  18Dec1900  14Jan2200  200  800  100  0.0014  0.27  3 
Metis  516  JUP310  18Dec1900  14JAN2200  200  600  100  0.0009  0.20  3 
Jovian Irregular Satellites
Sat.  Code  Ephem  Start Time  Stop Time  R (km)  T (km)  N (km)  P (s)  RA/Dec (arcsec)  Ref.  

Himalia  506  JUP340  06Feb1600  09Dec2599  200  300  200  6  0.10  4  
Elara  507  JUP340  06Feb1600  09Dec2599  200  400  200  10  0.13  4  
Pasiphae  508  JUP340  06Feb1600  09Dec2599  200  400  250  30  0.13  4  
Sinope  509  JUP340  06Feb1600  09Dec2599  250  700  300  40  0.23  4  
Lysithea  510  JUP340  06Feb1600  09Dec2599  300  800  400  15  0.28  4  
Carme  511  JUP340  06Feb1600  09Dec2599  300  900  300  50  0.30  4  
Ananke  512  JUP340  06Feb1600  09Dec2599  350  900  400  60  0.30  4  
Leda  513  JUP340  06Feb1600  09Dec2599  500  1200  400  50  0.40  4  
Callirrhoe  517  JUP340  06Feb1600  09Dec2599  tbd  tbd  tbd  tbd  Jovian Irregular Satellites (continued)
