2-3. Orbital Position and Velocity We obtain a scalar equation for the relative motion by substituting the expression for d2r/dt2 from Eq. (2.7) into Eq. (2.5); comparing the radial components gives (2.5) (2.7) (2.11) In order to find r as a function of , we need to make the substitution u=1/r and to eliminate the time by making use of the constant h=r2d/dt (Eq. 2.8). By differentiation r with respect to time, we obtain (2.12) and

and hence Eq. (2.11) can be written (2.13) Divided by ‘-u2h2’ The general solution of this second-order linear differential equation is (2.15) where e (an amplitude) and  (a phase) are two constant of integration, substituting back for r we have (2.14) which is the general equation of a conic in polar polar coordinates where e is the eccentricity (이심률) and p is the semi-latus rectum given by (2.16)

Four possible conic sections (원뿔 곡선) are: circle(원): e=0 p=a; ellipse (타원): 0<e<1 p=a(1-e2); parabola (포물선): e=1 p=2q; hyperbola (쌍곡선): e>1 p=a(e2-1); where the constant a is the semi-major axis of the conic. (2.17) The conic section curves derive their name from the curves formed by the intersection of various planes with the surface of a cone. The type of conic is determined by the angle of plane relative to the horizontal. If the plane is horizontal, that is, perpendicular to the axis of symmetry of the cone, then the resulting curve is a circle. If the angle is less than slope angle of the cone, then an ellipse results, whereas the angle is equal to the slope angle we have parabola. And it is beyond the slope angle, we have hyperbola results. Latus rectum Semi-latus rectum

Conic section in polar coordinate: Equation Eccentricity (e) Semi-latus rectum (l) Circle Ellipse Parabola Hyperbola Conic section in polar coordinate: Latus rectum (2l) Minor axis (2b) Major axis (2a)

The path of a planet around the Sun is elliptical in inertial space The path of a planet around the Sun is elliptical in inertial space. Note that the mass m1 is in focus of the ellipse and the other focus is empty. In the right figure, a is the semi-major axis, b semi-minor axis, e the eccentricity and  the longitude of pericenter. Since most objects in our Solar System are closed orbit, we focus on the elliptical motion in this lecture. In celestial mechanics it is customary to use the term of longitude when referring to any angle that is measured with respect to a reference line fixed in inertial space. The angle  is called the true longitude. Eq. (2.19) shows that the maximum and minimum values of the orbit radius are ra=a(1+e) and rp=a(1-e) when =+and =. These points are called apocentre (or aphelion) and pericentre (or perihelion), respectively. In the case of ellipse, there is a relationship between the quantities a, e, and b (2.18) We also have a relation from Eq. (2.15) and p=a(1-e2) (2.19)

It is convenient to refer the angular coordinate to the pericentre than to the arbitrary reference line. Therefore, we usually use the true anomaly defined as Hence Eq. (2.19) can be written (2.20) Using Cartesian coordinate system (직교 좌표계, see the Fig.), the components of the position vector are and (2.21) In one orbital period T, the area swept out by a radius vector is simply the area A=ab enclosed by the ellipse. From Eq. (2.10) this area has to equal hT/2 and hence, given that h2=a(1-e2), (2.22)

(2.22) which corresponds to Kepler’s third law of planetary motion. It is clear T is independent of e. Consider the case of two objects of mass m and m’, orbiting a object of mass mc. Let the orbiting object have semi-major axes a and a’ and orbital period T and T’. From Eq. (2.22) (2.23) If planet have a satellite, we can obtain the mass under the assumption of m’’<<m and m<<mc. (2.24) where mc, m and m’ are the mass of the Sun, planet and its satellite. This picture is an image of asteroid Ida, which was explored by Galileo mission. The semi-major axis and orbital period enable to determine the mass of the asteroid.

617 Patroclus Density = 0.8 g cm-3 (Marchis et al. 2006)

Since the angle  covers 2 radians in one period, we can define the “average” angular velocity, or the mean motion, n as (2.25) From Eq. (2.22) we can write and (2.26) By taking a scalar product of dr/dt and Eq. (2.5) we have or (2.27) Eq. (2.27) can be integrated with respect to time t (2.28)

(2.28) where is the square of the velocity. C is a constant of the motion. Eq. (2.28), often called the vis viva integral, shows that the orbital energy per unit mass is conserved. This means that the two-body problem has four constant of the motion, (1)the energy integral C (2) three components of the angular momentum integral, h. f  =0 Hence we derive the constant C. (2.29) Differentiating Eq. (2.10) with respect to t, =r (2.30)

Using Eq. (2.8) , Eq. (2.30) can be written (2.31) =r Eq. (2.20) =r (2.32) Then Eq. (2.29) can be written (2.33) (2.34) By comparing Eq. (2.34) with Eq. (2.28) we see that the energy constant can be written as Eq. (2.28) (2.35) and hence the energy of an elliptical orbit is a function of its semi-major axis alone and independent of the eccentricity. Similar quantities can be defined for parabolic and hyperbolic orbits. and (2.36)

Exercise Question: Consider the velocity at pericentre and apcentre of comet 17P/Holmes (a=3.6AU, e=0.4). (2.34) Since r=a(1-e) at pericentre and r=a(1+e) at apcentre,

From the conservation of energy 1/a (AU-1) N From the conservation of energy The right side, the energy constant for an elliptical orbit, is a function of its semi-major axis alone and is independent of the eccentricity. Similar quantities can be defined for parabolic and hyperbolic orbits. It can be shown that Cpara=0 and Chyper=/2a. Dones et al. (2004)

Oort cloud

For an orbital eccentricity of 0 For an orbital eccentricity of 0.9999 and a semimajor axis of 104 AU, the perihelion distance, q, and the aphelion distance, Q, are q=a(1-e)=1 AU Q=a(1+e)=19,999 AU, respectively. The orbital velocity at the perihelion is , compared to the local escape velocity at 1 AU, This means that a velocity increment of 1 m/s caused by a perturbation push the comet above the escape velocity of the Solar System.

Let us suppose that interstellar space is abundantly populated by comets that have escaped from the other stars. The mean velocity of these comets with respect to the Sun will be the same as the mean velocity of nearby stars relative to the Sun, about 20 km/s. where v=20km/s at r, and standard gravitational constant =1.3x1011(km3/s2). These “extrasolar” comets would have excess energies of -0.46. Note that the most energetic comet ever observed in Fig. 12 at -0.0007 on the 1/a scale. Therefore, it is quite unreasonable to attribute an of the observed comets to an extrasolar source.

Orbital Elements of Comets (Tp>200 years) Kreutz Sungrazers Figure 8.

Sun-grazing comet http://soho.esac.esa.int Figure 9.

Nowadays, the Kuiper belt is believed to be the main source for short-period comets. It is a region of the Solar System between 30 AU to ~55 AU from the Sun. The Kuiper belt is similar to the main-belt asteroid that consists of small bodies. The most widely-accepted hypothesis of its formation is that the Oort cloud's objects initially formed much closer to the Sun as part of the same process that formed the planets and asteroids, but that gravitational interaction with young gas giants such as Jupiter ejected them into extremely long elliptical or parabolic orbits

Figure 5. e-Q (aphelion distance) plot for short period comets. 2P/Encke P/Read (S3) Figure 5. e-Q (aphelion distance) plot for short period comets.

Figure 6. e-q (perihelion distance) plot for short period comets. P/Read (S3) 29P/Schwassmann-Wachamann 1 Main-Belt Figure 6. e-q (perihelion distance) plot for short period comets.

Jupiter-family comets Halley-type comets Jupiter-family comets Figure ７. i-Q (aphelion distance) plot for short period comets.