# Satellite/Planetary Orbits

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All planetary orbits contain 5 unusually stable points. These points are particularly important because they allow man-made 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 Earth-Sun 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 earth-sun 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 = semi-major 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 semi-major 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 high-precision ephemerides for the planetary satellites. Tables are grouped by planet. Table column headings are described below.

Martian Satellites

 Sat. Ref. Code Ephem Start Time Stop Time R(km) T(km) N(km) P(s) RA/Dec(arcsec) Phobos 401 MAR097 01-Jan-1600 04-Jan-2500 2 5 2 0.0001 0.01 1 Deimos 402 MAR097 01-Jan-1600 04-Jan-2500 3 10 3 0.0003 0.02 1

Galilean Satellites of Jupiter

 Sat. Ref. Code Ephem Start Time Stop Time R(km) T(km) N(km) P(s) RA/Dec(arcsec) Io 501 JUP310 18-Dec-1900 14-Jan-2200 5 5 5 0.0003 0.002 3 Europa 502 JUP310 18-Dec-1900 14-Jan-2200 5 5 5 0.0003 0.002 3 Ganymede 503 JUP310 18-Dec-1900 14-Jan-2200 5 5 5 0.0003 0.002 3 Callisto 504 JUP310 18-Dec-1900 14-Jan-2200 5 5 5 0.0003 0.002 3

Jovian Inner Satellites

 Sat. Ref. Code Ephem Start Time Stop Time R(km) T(km) N(km) P(s) RA/Dec(arcsec) Amalthea 505 JUP310 18-Dec-1900 14-Jan-2200 200 400 100 0.0003 0.13 3 Thebe 514 JUP310 18-Dec-1900 14-Jan-2200 200 400 100 0.0005 0.13 3 Adrastea 515 JUP310 18-Dec-1900 14-Jan-2200 200 800 100 0.0014 0.27 3 Metis 516 JUP310 18-Dec-1900 14-JAN-2200 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 JUP34006-Feb-1600 09-Dec-2599200 300200 60.10 4
Elara 507 JUP34006-Feb-1600 09-Dec-2599200 400200 100.13 4
Pasiphae 508 JUP34006-Feb-1600 09-Dec-2599200 400250 300.13 4
Sinope 509 JUP34006-Feb-1600 09-Dec-2599250 700300 400.23 4
Lysithea 510 JUP34006-Feb-1600 09-Dec-2599300 800400 150.28 4
Carme 511 JUP34006-Feb-1600 09-Dec-2599300 900300 500.30 4
Ananke 512 JUP34006-Feb-1600 09-Dec-2599350 900400 600.30 4
Leda 513 JUP34006-Feb-1600 09-Dec-2599500 1200400 500.40 4
Callirrhoe 517 JUP34006-Feb-1600 09-Dec-2599tbd tbdtbd tbd Jovian Irregular Satellites (continued)
Sat. Code Ephem Start Time Stop Time R
(km)
T
(km)
N
(km)
P
(s)
RA/Dec
(arcsec)
Ref.
S/2003 J 2 55060 JUP34006-Feb-1600 09-Dec-2599tbd tbdtbd tbd Saturnian Regular Satellites
 Sat. Ref. Code Ephem Start Time Stop Time R(km) T(km) N(km) P(s) RA/Dec(arcsec) Mimas 601 SAT389 25-Dec-1599 07-Jan-2600 5 50 5 tbd 0.009 5 Enceladus 602 SAT389 25-Dec-1599 07-Jan-2600 5 20 5 tbd 0.004 5 Tethys 603 SAT389 25-Dec-1599 07-Jan-2600 5 50 5 tbd 0.009 5 Dione 604 SAT389 25-Dec-1599 07-Jan-2600 5 20 5 tbd 0.004 5 Rhea 605 SAT389 25-Dec-1599 07-Jan-2600 5 20 5 tbd 0.004 5 Titan 606 SAT389 25-Dec-1599 07-Jan-2600 5 10 5 tbd 0.002 5 Hyperion 607 SAT389 25-Dec-1599 07-Jan-2600 10 100 5 tbd 0.017 5 Iapetus 608 SAT389 25-Dec-1599 07-Jan-2600 5 20 5 tbd 0.004 5 Methone 632 SAT393 26-Dec-1949 10-Jan-2050 tbd tbd tbd tbd tbd 6 Pallene 633 SAT393 26-Dec-1949 10-Jan-2050 tbd tbd tbd tbd tbd 6 Anthe 649 SAT393 26-Dec-1949 10-Jan-2050 tbd tbd tbd tbd tbd 6 Aegaeon 653 SAT393 26-Dec-1949 10-Jan-2050 tbd tbd tbd tbd tbd 6

Saturnian Librating Satellites

 Sat. Ref. Code Ephem Start Time Stop Time R(km) T(km) N(km) P(s) RA/Dec(arcsec) Helene 612 SAT389 25-Dec-1599 07-Jan-2600 5 20 5 tbd 0.004 5 Telesto 613 SAT393 26-Dec-1949 10-Jan-2050 5 75 5 tbd 0.013 6 Calypso 614 SAT393 26-Dec-1949 10-Jan-2050 5 75 5 tbd 0.013 6 Polydeuces 634 SAT393 26-Dec-1949 10-Jan-2050 tbd tbd tbd tbd tbd 6

Saturnian Inner Satellites

 Sat. Ref. Code Ephem Start Time Stop Time R(km) T(km) N(km) P(s) RA/Dec(arcsec) Janus 610 SAT393 26-Dec-1949 10-Jan-2050 10 50 5 tbd 0.01 6 Epimetheus 611 SAT393 26-Dec-1949 10-Jan-2050 10 50 5 tbd 0.01 6 Atlas 615 SAT393 26-Dec-1949 10-Jan-2050 10 200 5 tbd 0.04 6 Prometheus 616 SAT393 26-Dec-1949 10-Jan-2050 10 100 5 tbd 0.02 6 Pandora 617 SAT393 26-Dec-1949 10-Jan-2050 10 100 5 tbd 0.02 6 Pan 618 SAT393 26-Dec-1949 10-Jan-2050 tbd tbd tbd tbd tbd 6 Daphnis 635 SAT393 26-Dec-1949 10-Jan-2050 tbd tbd tbd tbd tbd 6

Saturnian Irregular Satellites

Sat. Code Ephem Start Time Stop Time R
(km)
T
(km)
N
(km)
P
(s)
RA/Dec
(arcsec)
Ref.
Phoebe 609 SAT38925-Dec-1599 07-Jan-260010 5010 tbd0.009 5
Ymir 619 SAT36820-Dec-1899 02-Jan-2100tbd tbdtbd tbd Saturnian Irregular Satellites (continued)
Sat. Code Ephem Start Time Stop Time R
(km)
T
(km)
N
(km)
P
(s)
RA/Dec
(arcsec)
Ref.
S/2004 S 7 65035 SAT36820-Dec-1899 02-Jan-2100tbd tbdtbd tbd Uranian Regular Satellites
 Sat. Ref. Code Ephem Start Time Stop Time R(km) T(km) N(km) P(s) RA/Dec(arcsec) Ariel 701 URA111 18-Dec-1899 06-Jan-2100 200 200 200 0.016 0.02 9 Umbriel 702 URA111 18-Dec-1899 06-Jan-2100 200 200 200 0.028 0.02 9 Titania 703 URA111 18-Dec-1899 06-Jan-2100 200 200 200 0.044 0.02 9 Oberon 704 URA111 18-Dec-1899 06-Jan-2100 200 200 200 0.100 0.02 9 Miranda 705 URA111 18-Dec-1899 06-Jan-2100 200 200 200 0.010 0.02 9

Uranian Inner Satellites

Sat. Code Ephem Start Time Stop Time R
(km)
T
(km)
N
(km)
P
(s)
RA/Dec
(arcsec)
Ref.
Cordelia 706 URA09101-Jan-1900 01-Jan-210050 14500200 0.151.1 11
Ophelia 707 URA09101-Jan-1900 01-Jan-2100500 75000250 0.95.7 11
Bianca 708 URA09101-Jan-1900 01-Jan-210030 410230 0.00600.03 11
Cressida 709 URA09101-Jan-1900 01-Jan-210025 220150 0.00350.02 11
Desdemona 710 URA09101-Jan-1900 01-Jan-210025 220150 0.00350.02 11
Juliet 711 URA09101-Jan-1900 01-Jan-210020 185150 0.00300.01 11
Portia 712 URA09101-Jan-1900 01-Jan-210025 175150 0.00300.01 11
Rosalind 713 URA09101-Jan-1900 01-Jan-210025 260180 0.00500.02 11
Belinda 714 URA09101-Jan-1900 01-Jan-210020 220150 0.00500.02 11
Puck 715 URA11118-Dec-1899 06-Jan-210020 200120 0.00500.02 9
Perdita 725 URA09101-Jan-1900 01-Jan-2100tbd tbdtbd tbd Uranian Irregular Satellites
Sat. Code Ephem Start Time Stop Time R
(km)
T
(km)
N
(km)
P
(s)
RA/Dec
(arcsec)
Ref.
Caliban 716 URA11230-Dec-1899 25-Dec-2099tbd tbdtbd tbd Neptunian Irregular Satellites
Sat. Code Ephem Start Time Stop Time R
(km)
T
(km)
N
(km)
P
(s)
RA/Dec
(arcsec)
Ref.
Triton 801 NEP08103-Dec-1599 01-Jan-26002 15050 tbd0.007 14
Nereid 802 NEP08103-Dec-1599 01-Jan-26001600 300060 tbd0.20 14
Halimede 809 NEP08625-Jan-1800 29-Nov-2199tbd tbdtbd tbd Neptunian Inner Satellites
Sat. Code Ephem Start Time Stop Time R
(km)
T
(km)
N
(km)
P
(s)
RA/Dec
(arcsec)
Ref.
Naiad 803 NEP08801-Jan-1900 04-Jan-210030 8560 0.01670.05 17
Thalassa 804 NEP08801-Jan-1900 04-Jan-210025 7040 0.00550.05 17
Despina 805 NEP08801-Jan-1900 04-Jan-210020 5030 0.00080.05 17
Galatea 806 NEP08801-Jan-1900 04-Jan-210015 3525 0.00100.05 17
Larissa 807 NEP08801-Jan-1900 04-Jan-210015 3525 0.00100.05 17
Proteus 808 NEP08130-Dec-1799 02-Jan-22004 25050 tbd0.01 14
S/2004 N 01814 NEP08801-Jan-1900 04-Jan-2100tbd tbdtbd tbd Pluto's Satellites
Sat. Code Ephem Start Time Stop Time R
(km)
T
(km)
N
(km)
P
(s)
RA/Dec
(arcsec)
Ref.
Charon 901 PLU05528-Dec-1599 04-Jan-260060 256 1.00.01 19
Nix 902 PLU05528-Dec-1599 04-Jan-2600110 1625210 tbd Table Column Headings
 Code The JPL satellite code number Ephem The JPL ephemeris identification number Start Time Ephemeris start time Stop Time Ephemeris stop time R Radial position uncertainty (km) T Downtrack position uncertainty (km) N Out-of-plane position uncertainty (km) P Orbital period uncertainty (s) RA/Dec Geocentric astrometric position uncertainty (arcsec)

### REFERENCES

2. Lieske, J.H. (1998) Galilean satellite ephemerides E5'', Astronomy & Astrophysics Supp. 129 , 205.

3. Jacobson, R.A. (2013) JUP310 - JPL satellite ephemeris.

4.Brozovic, M. and Jacobson, R. A. (2016) The Orbits of Jupiter's Irregular Satellites'', Astronomical Journal submitted .

5. Jacobson, R.A. (2016) SAT389 - JPL satellite ephemeris.

6. Jacobson, R.A. (2016) SAT393 - JPL satellite ephemeris.

7. Jacobson, R. A., Spitale, J., Porco, C. C., Beurle, K. Cooper, N. J., Evans, M. W., and Murray, C. D. (2008) Revised Orbits of Saturn's Small Inner Satellites'', Astronomical Journal 135 , 261.

8. Jacobson, R.A. (2014) SAT368 - JPL satellite ephemeris.

9. Jacobson, R. A. (2014) The Orbits of the Uranian Satellites and Rings, the Gravity Field of the Uranian System, and the Orientation of the Pole of Uranus'', Astronomical Journal 148 , 76.

10. Jacobson, R.A. (1998) The orbits of the inner Uranian satellites from Hubble Space Telescope and Voyager 2 observations'', Astronomical Journal 115 , 1195.

11. Brozovic, M. (2009) URA091 - JPL satellite ephemeris.

12.Brozovic, M. and Jacobson, R. A. (2009) The Orbits of the Outer Uranian Satellites'', Astronomical Journal 137 , 3834.

13.Jacobson, R. A. and Brozovic, M. (2014) URA112 - JPL satellite ephemeris.

14. Jacobson, R. A. (2009) The Orbits of the Neptunian Satellites and the Orientation of the Pole of Neptune'', Astronomical Journal 137 , 4322.

15.Brozovic, M., Jacobson, R. A., and Sheppard, S. S. (2011) The Orbits of the Outer Neptunian Satellites'', Astronomical Journal 141 , 135.

16. Jacobson, R. A. and Owen, Jr., W. M. (2004) The orbits of the inner Neptunian satellites from Voyager, Earthbased, and Hubble Space Telescope observations'', Astronomical Journal 128 , 1412.

17. Jacobson, R.A. (2015) NEP088 - JPL satellite ephemeris, an update to NEP050 (Jacobson and Owen 2004, AJ, 128.

18. Brozovic, M. and Jacobson, R. A. (2013) The Orbits, Masses of Pluto's Satellites'', Presented at The Pluto System on the Eve of Exploration by New Horizons: Perspectives, Predictions held at APL, Laurel, MD - PLU042 - JPL satellite ephemeris.

19. Jacobson, R. A. and Brozovic, M. and Buie, M. and Porter, S. and Showalter, M. and Spencer, J. and Stern, S. A. and Weaver, H. and Young, L. and Ennico, K. and Olkin, C., (2015) The Orbits and Masses of Pluto's Satellites after New Horizons'', Presented at the 47th AAS/Division for Planetary Sciences Meeting held at Gaylord Convention Center, National Harbor, MD - PLU055 - JPL satellite ephemeris.

20. Laskar, J. and Jacobson, R.A. (1987) GUST86. An analytical ephemeris of the Uranian satellites'', Astronomy & Astrophysics 188 , 212.

21. Jacobson, R.A., Riedel, J.E. and Taylor, A.H. (1991) The orbits of Triton and Nereid from spacecraft and Earthbased observations'', Astronomy & Astrophysics 247 , 565.

22. Owen, W.M., Vaughan, R.M, and Synnott, S.P. (1991) Orbits of the Six New Satellites of Neptune'', Astronomical Journal 101 , 1511.

23. Showalter, M.R. (1991) Visual detection of 1981S13, Saturn's eighteenth satellite, and its role in the Encke gap'', Nature 351 , 709.

24. Jacobson, R.A. (1998) The orbit of Phoebe from Earthbased and Voyager observations'', Astronomy & Astrophysics Supp. 128 , 7.

25. Jacobson, R.A. (2000) The orbits of the outer Jovian satellites'', Astronomical Journal 120 , 2679-2686.

26. Jacobson, R.A. (2004) The orbits of the major Saturnian satellites and the gravity field of Saturn from spacecraft and Earthbased observations'', Astronomical Journal 128 , 492.

27. Jacobson, R. A. and French, R. G. (2004) Orbits and Masses of Saturn's Coorbital and F-ring Shepherding Satellites'', Icarus 172 , 382.

28. Spitale, J. N., Jacobson, R. A., Porco, C. C., and Owen, Jr., W. M. (2006) The Orbits of Saturn's Small Satellites Derived from Combined Historic and Cassini Imaging Observations'', Astronomical Journal 132 , 692.

29. Showalter, M. R. and Lissauer, J. J. (2006) The Second Ring-Moon System of Uranus: Discovery and Dynamics'', Science 311 , 973.

30. Cooper, N. J., Murray, C. D., Evans, M. W., Beurle, K. Jacobson, R. A., and Porco, C. C. (2008) Astrometry and Dynamics of Anthe (S/2007 S 4), a New Satellite of Saturn'', Icarus 195 , 765.

## Satellite/Planetary Orbits - Astronomy

Let us define two unit vectors, and . (A unit vector is simply a vector whose length is unity.) As shown in Fig. 105, the radial unit vector always points from the Sun towards the instantaneous position of the planet. Moreover, the tangential unit vector is always normal to , in the direction of increasing . In Sect. 7.5, we demonstrated that when acceleration is written in terms of polar coordinates, it takes the form

These expressions are more complicated that the corresponding cartesian expressions because the unit vectors and change direction as the planet changes position.

Now, the planet is subject to a single force: i.e. , the force of gravitational attraction exerted by the Sun. In polar coordinates, this force takes a particularly simple form (which is why we are using polar coordinates):

The minus sign indicates that the force is directed towards, rather than away from, the Sun.

According to Newton's second law, the planet's equation of motion is written

The above four equations yield

where is a constant of the motion . What is the physical interpretation of ? Recall, from Sect. 9.2, that the angular momentum vector of a point particle can be written

For the case in hand, and [see Sect. 7.5]. Hence,

Clearly, represents the angular momentum (per unit mass) of our planet around the Sun. Angular momentum is conserved ( i.e. , is constant) because the force of gravitational attraction between the planet and the Sun exerts zero torque on the planet. (Recall, from Sect. 9, that torque is the rate of change of angular momentum.) The torque is zero because the gravitational force is radial in nature: i.e. , its line of action passes through the Sun, and so its associated lever arm is of length zero.

The quantity has another physical interpretation. Consider Fig. 106. Suppose that our planet moves from to in the short time interval . Here, represents the position of the Sun. The lines and are both approximately of length . Moreover, using simple trigonometry, the line is of length , where is the small angle through which the line joining the Sun and the planet rotates in the time interval . The area of the triangle is approximately

i.e. , half its base times its height. Of course, this area represents the area swept out by the line joining the Sun and the planet in the time interval . Hence, the rate at which this area is swept is given by

Clearly, the fact that is a constant of the motion implies that the line joining the planet and the Sun sweeps out area at a constant rate : i.e. , the line sweeps equal areas in equal time intervals. But, this is just Kepler's second law. We conclude that Kepler's second law of planetary motion is a direct manifestation of angular momentum conservation .

where is a new radial variable. Differentiating with respect to , we obtain

The last step follows from the fact that . Differentiating a second time with respect to , we obtain

Equations (567) and (578) can be combined to give

This equation possesses the fairly obvious general solution

where and are arbitrary constants.

The above formula can be inverted to give the following simple orbit equation for our planet:

The constant merely determines the orientation of the orbit. Since we are only interested in the orbit's shape , we can set this quantity to zero without loss of generality. Hence, our orbit equation reduces to

Formula (582) is the standard equation of an ellipse (assuming ), with the origin at a focus. Hence, we have now proved Kepler's first law of planetary motion. It is clear that is the radial distance at . The radial distance at is written

Here, is termed the perihelion distance ( i.e. , the closest distance to the Sun) and is termed the aphelion distance ( i.e. , the furthest distance from the Sun). The quantity

is termed the eccentricity of the orbit, and is a measure of its departure from circularity. Thus, corresponds to a purely circular orbit, whereas corresponds to a highly elongated orbit. As specified in Tab. 7, the orbital eccentricities of all of the planets (except Mercury) are fairly small.

According to Eq. (575), a line joining the Sun and an orbiting planet sweeps area at the constant rate . Let be the planet's orbital period. We expect the line to sweep out the whole area of the ellipse enclosed by the planet's orbit in the time interval . Since the area of an ellipse is , where and are the semi-major and semi-minor axes, we can write

Incidentally, Fig. 107 illustrates the relationship between the aphelion distance, the perihelion distance, and the semi-major and semi-minor axes of a planetary orbit. It is clear, from the figure, that the semi-major axis is just the mean of the aphelion and perihelion distances: i.e. ,

Thus, is essentially the planet's mean distance from the Sun. Finally, the relationship between , , and the eccentricity, , is given by the well-known formula

This formula can easily be obtained from Eq. (582).

Equations (584), (585), and (588) can be combined to give

It follows, from Eqs. (587), (589), and (590), that the orbital period can be written

Thus, the orbital period of a planet is proportional to its mean distance from the Sun to the power --the constant of proportionality being the same for all planets. Of course, this is just Kepler's third law of planetary motion.

## Satellite/Planetary Orbits - Astronomy

Find someone who knows:

1) The names of the planets in order.

2) Three differences between the terrestrial planets and the gas giants.

3) The largest planet in the solar system.

4) The hottest planet in the solar system.

5) A fact about a planet in our solar system.

6) A fact about a moon in our solar system.

7) What comets, asteroids, satellites and galaxies are.

Your teacher will tell you which questions to discuss.

Applet credit: this excellent Newton's Mountain applet was written by Michael Fowler , University of Virginia.

See more fantastic Fowler's Physics Applets here .

This thought experiment is sometimes referred to as "Newton's Cannonball".

Isaac Newton imagined firing a cannonball from a tall mountain above the earth's atmosphere, in his famous thought experiment Newton's Mountain*.

Try firing the cannonball at different velocities and see what happens!

Applet credit: Michael Fowler , University of Virginia.

Click the image below to load the applet from his site.

*Your teacher may explain some of Isaac Newton's ideas to you.

These animations show the paths that would be followed by objects launched at different speeds.

Click on any of the animations to load in full screen.

By Lookang many thanks to author of original simulation = Todd Timberlake author of Easy Java Simulation = Francisco Esquembre - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=21961928

Task 3a - Orbits of planets, moons and comets

See more fantastic Flash, Java and HTML5 simulations FREE at http://phet.colorado.edu

Note: PC only - this applet does not run on tablets or mobile devices.

This is another fun "orbits" applet, which runs on most platforms (Windows, Mac OS X and Linux).

Install Java to run this simulation (requires a proper computer - tablets and mobiles cannot run Java)

Use the orbital simulator to investigate the orbits of planets, moons and comets .

Applet credit: PhET . Load the simulation in full screen mode for best results.

Try the presets "Sun, planet", "Sun, planet, moon", and "sun, planet, comet".

You could also try setting up your own orbital systems!

Set the slider to "accurate" for higher precision calculations and better rendering of orbits. Switch off traces and observe the orbit carefully, and then switch them on to find out more about the orbital dynamics! Challenge: Try to set a moon in a retrograde orbit!

Click the image below to open the simulation in full screen mode*.

## A Tale of Planetary Resurrection

Shortly after NASA's Kepler mission began operations back in 2009, it identified what was thought to be a planet about the size of Neptune. Called KOI-5Ab, the planet, which was the second new planet candidate to be found by the mission, was ultimately forgotten as Kepler racked up more and more planet discoveries. By the end of its mission in 2018, Kepler had discovered a whopping 2,394 exoplanets, or planets orbiting stars beyond our sun, and an additional 2,366 exoplanet candidates, including KOI-5Ab.

Now, David Ciardi, chief scientist of NASA's Exoplanet Science Institute (NExScI), located at Caltech's IPAC, says he has "resurrected KOI-5Ab from the dead," thanks to new observations from NASA's TESS (Transiting Exoplanet Survey Satellite) mission.

"KOI-5Ab fell off the table and was forgotten," says Ciardi, who presented the findings at a virtual meeting of the American Astronomical Society (AAS). By 2014, Ciardi and other researchers had used the W. M. Keck Observatory in Hawaii, Caltech's Palomar Observatory near San Diego, and Gemini North in Hawaii to show that the star circled by KOI-5Ab is one member of a triple-star system called KOI-5. But they were not sure if the KOI-5 system actually hosted a planet or if they were seeing an erroneous signal from one of the two other stars.

Then, in 2018, TESS came along. Like Kepler, TESS looks for the blinking of starlight that comes when a planet crosses in front of, or transits, a star. TESS observed a portion of Kepler's field of view, including the KOI-5 system. Sure enough, TESS also identified KOI-5Ab as a candidate planet (though TESS calls it TOI-1241b). TESS, like Kepler, found that the planet orbited its star roughly every five days. But at that point, it was still not clear if the planet was real.

"I thought to myself, 'I remember this target,'" says Ciardi, after seeing the TESS data. He then went back and reanalyzed all the data, including that from the California Planet Search, led by Caltech professor of astronomy Andrew Howard. The California Planet Search uses ground-based telescopes, including the Keck Observatory, to search for the telltale wobble in a star that occurs when a planet circles around it and exerts a gravitational tug.

"If it weren't for TESS looking at the planet again, I would never have gone back and done all this detective work," says Ciardi.

Jessie Dotson, the Kepler/K2 project scientist at NASA Ames Research Center, says, "This research emphasizes the importance of NASA's full fleet of space telescopes and their synergy with ground-based systems. Discoveries like this one can be a long haul."

Together, the data from the space- and ground-based telescopes helped confirm that KOI-5Ab is a planet. KOI-5Ab is about one half the mass of Saturn and orbits a star (star A) with a relatively close companion (star B). Star A and star B orbit each other every 30 years. A third gravitationally bound star (star C) orbits stars A and B every 400 years.

The combined data set also reveals that the orbital plane of the planet is not aligned with the orbital plane of the second inner star (star B) as might be expected if the stars and planet all formed from the same disk of swirling material. Triple-star systems, which make up about 10 percent of all star systems, are thought to form when three stars are born together out of the same disk of gas and dust.

Astronomers are not sure what caused the misalignment of KOI-5Ab but speculate that the second star gravitationally kicked the planet during its development, skewing its orbit and causing it to migrate inward.

This is not the first evidence for planets in double- and triple-star systems. One striking case involves the triple-star system GW Orionis, in which a planet-forming disk had been torn into distinct misaligned rings, where planets may be forming. Yet despite hundreds of discoveries of multiple-star system planets, the frequency of planet formation in these systems is lower than that of single-star systems. This could be due to an observational bias (single-star planets are easier to detect) or because planet formation is in fact less common in multiple-star systems.

Future instruments, such as the Palomar Radial Velocity Instrument (PARVI) at the 200-inch Hale Telescope at Palomar and the Keck Planet Finder at Keck, will open up new avenues for better answering these questions.

"Stellar companions may partially quench the process of planet formation," says Ciardi. "We still have a lot of questions about how and when planets can form in multiple-star systems and how their properties compare to planets in single-star systems. By studying the KOI-5 system in more detail, perhaps we can gain insight into how the universe makes planets."

## Satellite Orbits

Many satellites move in stable orbits around the earth, some have left the gravitational field of the Earth and move around other planets and a couple are on the way to leave the solar system. An important task of astronomy is therefore to determine the orbits of these satellites.

### Trajectories

In the calculation of satellite orbits two speeds ( v_1, v_2 ) play a major role. They depend on the starting altitude of the satellite (see below for derivation).

The first velocity ( v_1 ) is the launching speed, which ensures a stable circular orbit. If the satellite is launched at a higher speed its orbit is deformed into an ellipse. If a satellite, however, is launched with a smaller velocity it will probably fall to the ground and explode.

The second velocity ( v_2 ) is the launching speed, which empowers a satellite to escape the gravitational field of the Earth on a parabolic path. If a satellite is launched with even higher speed it will leave the gravitational field on a hyperbolic path.

## Satellite/Planetary Orbits - Astronomy

Among the many things that NASA engineers consider when designing a satellite is its orbit, including which one is best for the data it will collect and how much maneuvering it will take to keep it there. Throughout the design process, engineers make calculations using the same laws of physics that were developed to explain the orbits of planets. This series of articles details the development of the science of orbital mechanics, catalogs the most common orbits of Earth-observing satellites, and shadows the engineers in mission control as they work to keep a satellite in orbit.

Few ideas have had a greater impact on humanity than our quest to understand why things orbit across the heavens. Planetary Motion: The History of an Idea That Launched a Scientific Revolution describes how the study of the motion of the planets led to the development of the basic theories of motion and gravity that are used to calculate a satellite&rsquos orbit.

Satellites are designed to orbit Earth in one of three basic orbits defined by their distance from the planet. Within these three orbits are many variations, each intended to provide the best view of Earth for the type of information the satellite is collecting. Catalog of Earth Satellite Orbits describes the most common orbits for Earth-observing satellites.

Finally, in Flying Steady: Mission Control Tunes Up Aqua&rsquos Orbit, the Earth Observatory peeks in on the Earth Observing System Mission Control Center as flight engineers adjust the path of NASA&rsquos Aqua satellite to keep it in the proper orbit for collecting scientific data.

## 10 New Alien Planets a Diverse Bunch, Telescope Shows

A European space telescope has discovered 10 previously unknown alien planets, including two Neptune-like objects that circle the same star, researchers announced today (June 14).

France's CoRoT satellite detected the 10 alien planets, all of which are gaseous like Saturn or Jupiter. However, they exhibit a range of masses, densities, orbital characteristics and other properties, researchers said. The new discoveries highlight the diversity of worlds beyond our solar system and boost the confirmed count of extrasolar planets up to 565, they added.

"Ever since the early days of exoplanet astronomy, we&rsquove been amazed by the variety of planets that have been discovered: gaseous giants larger than Jupiter and smaller, rocky bodies, down to masses comparable to the Earth&rsquos," said Malcolm Fridlund, the European Space Agency's project scientist for CoRoT, in a statement. [Photos: The Strangest Alien Planets]

Researchers announced the findings today (June 14), at the Second CoRoT Symposium in Marseille, France.

Alien planet haul

Like NASA's Kepler Space Telescope, CoRoT searches for alien planets by what is known as the transit method. This technique looks for tiny dips in a star's brightness that could potentially be caused by a planet passing in front of it from our perspective.

The 10 newly discovered alien worlds have all been confirmed by follow-up observations using ground-based telescopes. Seven of the discoveries are so-called "hot Jupiters," gas giants that orbit extremely close to their parent stars. Another one is smaller than Saturn, and the other two are Neptune-like siblings circling the same star.

While all the newfound alien planets are gaseous, they make up a diverse group. Their densities, for example, span a wide range, from values similar to that of Saturn (the least dense planet in our solar system) to densities comparable to that of rocky Mars, researchers said.

One planet orbits a 10-billion-year-old star, which is twice as old as the sun. Another circles a star just 600 million years old. Two of the exoplanets also lie on highly elongated orbits &mdash a surprise to scientists, considering how unstable such paths are thought to be.

A planetary zoo

Since the first planet beyond our solar system was discovered back in the 1990s, astronomers have discovered an astonishing diversity of alien worlds.

"The new set of 10 planets that we announce today are no exception, exhibiting as they do a rich list of very interesting properties," Fridlund said.

To date, astronomers have confirmed at least 565 alien planets, and the Kepler project has already identified 1,235 more "candidate" planets that await in-depth follow-up study. Researchers have predicted that at least 80 percent of Kepler's planetary candidates will eventually be confirmed.

Since its launch in 2006, CoRoT has detected several hundred candidate planet-hosting stars. The 10 new finds bring the satellite's total number of confirmed planet discoveries to 26.

Many more finds will likely follow &mdash from Kepler, CoRoT and other instruments &mdash helping astronomers better understand alien planets on a broader scale, researchers said.

"Although the study of exoplanets is relatively young, we have already reached a stage where we can characterize the details of worlds orbiting other stars, and CoRoT is making a crucial contribution to this field," Fridlund said. "With hundreds of systems observed to date, we no longer have to worry about 'taming the beasts' and we can dedicate our efforts to the 'zoology' of exoplanets, which is enormously enhancing our knowledge about planetary systems."

## Notes

[1] On this flight, SpaceX is also testing an experimental darkening treatment on one satellite to further reduce the albedo of the body of the satellites.

[3] Wood, Lloyd, Satellite constellation networks, Internetworking and Computing over Satellite Networks. Springer, Boston, MA, 2003, p.13-34.

[4] Such ground-based observatories are essential complements to astronomical satellites. Costs and limitations on size and weight preclude the launch of particularly large telescopes, and the difficulty of repairing and maintaining telescopes in space means that the newest, most revolutionary technologies are implemented on ground-based telescopes decades before it is prudent to attempt them in space. Both space-based and ground-based telescopes are fundamental to astronomy.

[5] The Vera C. Rubin Observatory will have the ability to survey the entire sky every three nights.