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Solar Nebula •  The solar nebula condensed to form small planetesimals •  Inner solar system: over 200 Kelvin, only metal and rock condense. •  Outer solar system: under 200 Kelvin, ice condenses as well. •  As the solar nebula cooled, material condensed to form planetesimals a few km across. The planetesimals collided to form larger planets. •  Terrestrial planets are rock and metal: They formed in hot inner region. •  Jovian planets include ice, H, He: They formed in cool outer region. •  Hydrogen and helium and gaseous everywhere. •  Not every planetesimal merged to form a planet. Comets = leAover icy planetesimals. Asteroids = leAover rocky and metallic planetesimals. EllipDcal Orbits and Orbital Elements Kepler’s Laws •  (A): Planets move in ellipEcal orbits with the Sun at one focus of the ellipse (not the center!). •  (B): During their orbits around the Sun, the planets sweep out equal areas in equal Emes (in other words, they move faster when they are closer to the Sun). •  (C): The period (P, in years), which is the Eme for a planet to orbit around the Sun, and semi-­‐major axis (a, in Astronomical Units) of a planet's orbit saEsfy the equaEon: P2 = a3 •  The orbits of the planets are ellipses. The sun seats at one focus of the ellipEcal orbit. In pracEce, the orbits are close to being circular, but not quite: the difference is noEceable. Let’s have a look at the main features of an ellipEcal orbit b a distance of closest approach greatest distance r (aphelion) (perihelion) •  Planets move in ellipses, is the statement of Kepler’s first law. The geometrical orbital elements are: –  a the semi-­‐major axis of the ellipse, which establishes the scale of the orbit –  e the eccentricity (elongaEon of the ellipse), which establishes the shape –  i the inclinaEon, which establishes the Elt of the orbital plane, relaEve to the eclipEc plane –  Ω the longitude of the ascending node (ascending crossover point of the orbit, measured with respect to the line to the vernal equinox), which establishes the orientaEon of the orbital plane –  ω the argument of perihelion (measured from the ascending node to the perihelion point; the sense is in the direcEon of orbital moEon), which establishes the orientaEon of the ellipse within the orbital plane –  ν the true anomaly, measured from perihelion point to object’s current posiEon –  T0 the epoch, which in this context, is an instant when the object is at perihelion •  In an ellipEcal orbit, the distance “r ” from the Sun (an object at the focus of the ellipEcal orbit) to the orbiEng planet (or a body) at any given Eme is: r = rp(1 + e)/[1 + e cosv] rp = a(1 – e), perihelion ra = a(1 + e), aphelion r = a(1 – e2)/[1 + e cosv] The Two-­‐Body Problem •  We will start by using Newton’s Laws of moEon to study the two-­‐
body problem. Newton’s Laws •  Law I: an object remains at rest or moves in a straight line in a uniform moEon unless an external force is applied to it. •  Law II: the acceleraEon “ac” of an object is proporEonal to the net force (the sum of all forces) applied on the object and inversely proporEonal to the mass “m” of the object. F = mac, •  Law III: If two objects exert forces on each other, these forces are equal in magnitude and opposite in direcEon, ie, F12 = -­‐ F21, where F12 = the force on object 1 exerted by object 2. •  Given two objects of mass m1 and m2, we want to determine the orbit of either one around the other, assuming that the only force acEng is their mutual gravitaEonal abracEon. Newton’s law of gravitaEon Two masses in an inerEal coordinate system •  General expression for the force exerted on an object of mass m, by another object with mass M: –  These could be two stars that orbit one another, –  or a planet in orbit about one star, –  or two unbound bodies that encounter each other •  Now we will us both Newton’s and Kepler’s laws to explain the orbital moEons of the planets Kepler’s Laws •  (A): Planets move in ellipEcal orbits with the Sun at one focus of the ellipse (not the center!). •  (B): During their orbits around the Sun, the planets sweep out equal areas in equal Emes (in other words, they move faster when they are closer to the Sun). •  (C): The period (P, in years), which is the Eme for a planet to orbit around the Sun, and semi-­‐major axis (a, in astronomical units) of a planet's orbit saEsfy the equaEon P2=a3 • 
Newton's Laws of moEon and gravitaEon could be used to derive Kepler's Laws (if we consider a circular orbit with a radius r) mM
v2
GM
F = ma c = G 2 = m
, here also we can write ac = 2
r
r
r
and
4π 2r 2
v =
P2
2
if we insert the definition of v in force equation then,
2π r
P=
⇒
v
2
4π 2r 2
2
mM
P
G 2 =m
r
r
⇒
2
4
π
P2 =
r3
GM
•  The more general definiEon of Kepler’s third law for an ellipEcal orbit: 2
2
P
4π
=
3
G(M + m)
a in this equaEon “P” in years, “a” in astronomical unit (AU) and “M” and “m” are in solar masses! €
•  Example: An object is 4 AU far from the Sun and its mass is 1/10th the Sun's. Calculate its orbital period? (≈ 7.6 years) •  Example: Calculate the orbital period of the Earth. (m⊕ = 5.97x1027g, a=1.49x108 km, M¤= 2x1033 g, G=6.67x10-­‐8 cm3g-­‐1s-­‐2) •  Example: Calculate the gravitaEonal acceleraEon on Earth by using the Newton’s moEon and gravitaEonal laws (R⊕ = 6.38x108 cm) •  Exercise: Calculate the orbital period of Moon. Assuming that it has a circular orbit, determine its orbital speed and the acceleraEon needed to keep the Moon on its orbit and compare it to the acceleraEon (again on Moon’s orbit) due to Earth’s gravity.