Ministry of Education of the Republic of Azerbaijan İndividual work

Ministry of Education of the Republic of Azerbaijan
İndividual work
Faculty/Institute: Pedagogy
Speciality(ies)/Course(s): Physics teacher

Subject: Mathematical Methods of Physics

Topic: Orbit of comets: basic equations, orbit program.
Lecturer: Prof Veli Huseynov
Student: Nigar Musayeva
Course: II
Education year: 2021-2025

ORBITS OF COMETS
Basic Equations
Consider the Kepler problem in which a small satellite, such as a comet, orbits the Sun. Wc use a Copernican coordinate system and fix the Sun at the origin. For now. consider only the gravitational force between the comet and the Sun, and neglect all other forces (e.g., forces due to the planets, solar wind). The force on the comet is
(3.1)
where r is the position of the comet, m is its mass.
M (= 1.99 x 10³⁰ kg) is the mass of the Sun, and
G (= 6.67 x 10^-11 m³/kg*s²) is the gravitational constant.
The natural units of length and time for this problem are not meters and seconds. As a unit of distance we will use the astronomical unit (AU; 1 AU = 1.496 x 10¹¹ m), which equals the mean Earth-Sun distance. The unit of time will be the AU year (the period of a circular orbit of radius 1 AU). In these units, the product GM= 4 ²AU³/yr². We take the mass of the comet, m, as unity; in MKS units the typical mass of a comet is 10^{15 3} kg.
We now have enough to assemble our program, but before doing so let’s quickly review what we know about orbits. For a complete treatment, see any of the standard mechanics texts, such as Symon or Landau and Lifshitz. The total energy of the satellite is
(3.2)
where r = |r| and v = |v|. This total energy is conserved, as is the angular momentum,
(3.3)
Since this problem is two-dimensional, we will take the motion to be in the xy plane. The only nonzero component of the angular momentum is in the z-direction.
When the orbit is circular, the centripetal force is supplied by the gravita­tional force,
(3.4)
or
(3.5)
To put in some values, in a circular orbit at r = 1 AU the orbital speed is v = 2 AU/yr (about 30,000 km/h). Using (3.5) in (3.2), the total energy in a circular orbit is
(3.6)
In an elliptical orbit, the semimajor and semiminor axes, a and b. are unequal (Figure 3.1). The eccentricity, e, is defined as
(3.7)
Earth’s eccentricity is e = 0.017, thus its orbit is nearly circular. The distance from the Sun at perihelion (closest approach) is q = (1-e)a; the distance from the Sun at aphelion is Q = (1 + e)a.
Equation (3.6) also holds for elliptical orbits if we replace the radius with the semimajor axis; that is, the total energy is
(3.8)
Note that E 0. From (3.2) and (3.8), we find that the orbital speed as a function of radial distance is
(3.9)

Figure 3.1: Elliptical orbit about the Sun.


Table 3.1: Orbital data for selected comets

Comet name	T(yrs)	e	q(AU)	i	First pass
Encke	3.30	0.847	0.339	12.4°	1786
Biela	6.62	0.756	0.861	12.6°	1772
Schwassmann-Wachmann 1	16.10	0.132	5.540	9.5°	1925
Halley	76.03	0.967	0.587	162.2°	239 B.C.
Grigg-Mellish	164.3	0.969	0.923	109.8°	1742
Halc-Bopp	2508	0.995	0.913	89.4°	1995

The speed is maximum at. perihelion and minimum at aphelion, the ratio of these speeds being Q/q. Finally, using conservation of angular momentum, we may derive Kepler’s third law,
(3.10)
where T is the period of the orbit.The orbital data for a few well-known comets are given in Table 3.1. The inclination, is the angle between the orbital plane of the comet and the ecliptic plane (the plane of the orbit of the planets). When the inclination is less than 90°, the orbit is said to be direct, when it is greater than 90°, the orbit is retrograde (i.e., orbits the Sun in the opposite direction from the planets).
Orbit Program
A simple program, called o rb it, that computes orbits for the Kepler problem using various numerical methods is outlined in Table 3.2. The Euler method, described in Section 2.1, computes the comet’s trajectory as
(3.11)
Table 3.2: Outline of program o rb it, which computes the trajectory of a comet using various numerical methods.

Set initial position and velocity of the comet.

Set physical parameters (m. GM, etc.)

Loop over desired number of steps using specified numerical method. - Record position and energy for plotting. - Calculate new position and velocity using: * Euler method (3.11), (3.12) or; * Euler-Cromer method (3.13), (3.14) or; * Fourth-order Runge-Kutta method (3.28), (3.29) or; * Adaptive Runge-Kutta method.

Graph the trajectory of the comet.

Graph the energy of the comet versus time.

where a is the gravitational acceleration. Again, we discretize in time and use the notation where is the time step.
The simplest test case is a circular orbit. For an orbital radius of 1 AU, Equation (3.5) gives a tangential velocity of 2 AU/yr. Fifty data points per orbital revolution should give us a smooth curve, so = 0.02 yr (or about one week) is a reasonable time step. With these values, the o rb it program, using the Euler method, gives the results shown in Figure 3.2. We immediately see that the orbit is not a circle, but an outward spiral. The reason is clear from the energy graph; instead of being constant, the total energy is continuously increasing. This type of instability is also observed in the Euler method for the simple pendulum (see Section 2.2).
Fortunately, there is a simple solution to this problem—the Euler-Cromer method computes the trajectory as
(3.13)
(3.14)