HenrykDot

★ HenrykDot.com ★
is the online companion to a series of books published by AIUT under the common main title
"Physics of My Imaginary Space-Time" by Henryk Dot.

POLSKI

last update
01/02/2023

Physics 3 - Return
Return to Physics 3

Physics 4 -
Home
Introduction

Physics 4 - Chapter 1
Motion
Orbital motion

Books published by AIUT
are found in libraries according to the list of compulsory copies.

Second Edition of "Fizyka 3"
ISBN 978-83-926856-1-6
Fizyka

can be bought in Warsaw
in the Academic Bookstore
PW Publishing House
Noakowskiego street 18/20

and in Katowice
in the bookstore "Liber"
Bankowa street 11.
(area of Silesian University)

English edition of "Physics"
ISBN 978-83-926856-2-3
Fizyka

is also in libraries
and the distribution method should be asked wydawca@aiut.com.

2. Orbital Motion Considerations on orbital motion will come useful for further sections of the book. The motion of a single object will be considered. In a real world, there are multiple objects interacting with each other. Discussion of a single object is sufficient to present the essence of the issue. Perhaps many graduate students will find this presented approach as preferable way to deal with this topic. We will consider motion in which the acceleration is described by the relation:
	(1.18)
where: r ⃗ - position vector, r - position vector length (modulus), - functional relation: acceleration dependence on distance.
In our considerations, b will take constant values, as for example in the case of gravity b = k_g ⋅M or for case of atom b = Z⋅ e² / (4⋅π⋅m_e⋅ ε₀) where: k_g - gravitational constant, M - mass of the central object, Z - number of protons in a nucleus, e - electron charge, m_e - electron mass, ε₀ - vaccum permittivity.
Many of consideration/conclusions are still correct when b is not a constant but a function of r i.e., b = b(r). For the sake of clarity, we begin our considerations with an assumption that the central object is stationary and its interaction is immediate. This has no significant impact on the validity of the conclusions drawn at this stage. In some problems, a consideration of the motion of the central object and the delay of its interaction will be very important, but we will deal with these in another section. Orbital motion is conveniently analyzed in polar coordinates r and α such as:
x = r ⋅ cos(α). ,	(1.19)
y = r ⋅ sin(α) .	(1.20)
We obtain then the following relations
,	(1.21)
,	(1.22)
,	(1.23)
,	(1.24)
Inserting (1.23) and (1.24) respectively into (1.18) we obtain:
,	(1.25)
and
.	(1.26)
Now, performing multiplications (1.25) and (1.26) by cos(α) and sin(α) respectively, then adding, and then subtracting both equations by sides and finally replacing the expression with new variable we obtain very clear and simple formulas (1.27) and (1.28).
	(1.27)
	(1.28)
We can also rewrite formula (1.27) by expressing ω by , where v_s - circumferential velocity or in other words tangential velocity to a circle with radius r
.	(1.29)
It should be noted that formula (1.28) does not use any variables besides r and ω and which is very important for farther considerations it does not contain the value b ! We just obtained a complete set of orbital formulas (1.27) and (1.28). Very often instead of these two, only one formula is presented applicable to circular orbit:
	(1.30)
It represents the specific case of formula (1.29), when . However, formula (1.30) still can be useful for us. When we rewrite it in the form as below,
	(1.31)
it can be noticed that (using interesting relation from formula (1.17)) an object in orbital motion on circular orbit contains kinetic energy equal to a half of the energy required to leave a field. The velocity momentum is constant not only for orbital motion. If an object moves in a straight line with a constant speed, the constant velocity momentum is conserved in respect to any point lying outside the straight line of motion. The principle of conservation of angular momentum is directly related to the principle of conservation of velocity momentum. Formula (1.28) can be written as:
	(1.32)
which results in
	(1.33)
and after substitution of ω by
.	(1.34)
It is the same as
	(1.35)
thus
	(1.36)
and this allows us to express our desired rule without using a mass m.
r ⋅ v_s = k = const	(1.37)
*Obviously, since r ⋅ v_s = k = const* thus m ⋅ v_s ⋅ r must be a constant value as well.** Formula (1.37) is equivalent to its three-dimensional vector representation:
r ⃗ × v ⃗ = k ⃗	(1.38)
Relation shown in (1.37) can be used as constant in the formula (1.29), which then be expressed as:
	(1.39)
If k and b are constant, the expression is also constant and it determines certain radius, which we can denote as and then express formula (1.39) as:
	(1.40)
This formula is very interesting, and together with (1.37) encompasses a different approach in presenting the property known as the Laplace-Runge-Lenz vector.
Let us try to find some interesting features of equation (1.40). Let v_r be denoted as a radial component of the velocity
	(1.41)
We can then use the following relation
	(1.42)
and express equation (1.40) as:
	(1.43)
In this equation, the variables are conveniently separated, so we can integrate on both sides of the equation, yielding:
	(1.44)
where C is an integration constant. We can also eliminate the ½ coefficient on the left side of the equation:
	(1.45)
To determine integration constant C we will take advantage of the fact that at certain extreme points r, radial speed equals zero. We can write the equation for these points:
	(1.46)
which leads to a quadratic equation:
0 = - b ⋅ r_k + 2⋅b ⋅ r + C ⋅ r²	(1.47)
The equation has two solutions:
	(1.48a)
and
	(1.48b)
Denoting:
	(1.49)
formula(1.48) can be written as:
	(1.50)
and
We easily notice that
	(1.51)
and
.	(1.52)
In addition (1.51) represents the sum of the distances from any point on the perimeter to both foci of the ellipse which are used to plot the ellipse. Thus equation (1.45) can be expressed as:
	(1.53)
and then including (1.51) and (1.52)
	(1.54)
The same result can be yielded when using energy formulas.
	(1.55)
where: E – total energy, v – total speed. For orbital motion, the total energy of such system is negative. Calculations will be easier to conduct when we relate the energy to a unit of mass and, we denote it with the index m i.e.
	(1.56)
By substituting total speed by its components, we obtain:
	(1.57)
which is equivalent to formula (1.45) best visible after equation expressions are rearranged
	(1.58)
and it is the same equation since: k² = b ⋅ r_k and 2 ⋅ E_m = C . This prompted a suggestion to associate radius r with energy and assign the index E to it. When deriving the formula for v_r relation (1.55) was not used to avoid the use of the terms work and energy, which we had not defined before. They are referenced only to demonstrate full compatibility of mathematical derivations. The new values introduced as radii r_k and r_E seem to better reflect the parameters of the ellipse than the popularly used term– eccentricity.
Then the equation of an ellipse using r_E and r_k values:
	(1.59)
and in parametric form, which is easy to plot:
	(1.60a)
	(1.60b)
It is useful to remember the characteristic values:
longer semi-axis of an ellipse r_E ,	(1.61)
shorter semi-axis ,	(1.61a)
amplitude of a radius changes .	(1.61b)
Amplitude of a radius changes corresponds to the distance of the foci of the ellipse from the center of the ellipse and is directly related to commonly used concept - eccentricity. The characteristic quantities can be supplemented with the time of the ellipse circulation, which in our case represents the period of radius change i.e. the transition time from the maximum radius to the minimum radius and back to the maximum radius. To obtain this value, it is sufficient to calculate half of this period that is, the transition time from the minimum to the maximum radius. We will use the definition of radial velocity
.	(1.62)
Thus,
and	(1.63)
Substituting v_r with formula (1.54) we obtain (1.54)
	(1.64)
then, after solving the integral this equals to:
and results with the answer
.	(1.65)
From formula (1.65) we derive Kepler’s Third Law
.	(1.66)
The description of orbital motion proposed here using two parameters, r_E i r_k characterizes the properties of orbital motion very well. Keeping either parameter r_E or r_k constant, we obtain two families of orbits shown in the figures below, one with constant energy and the other with constant velocity momentum (Fig. 2). The r_E radius corresponds to the energy and also determines the radius against which the value varies with amplitude . The radius r_k determines the radius with respect to which the direction of the force changes along the radius.. For r > r_k here is an attraction, and for r < r_k a repulsion. For an ellipse family of constant energy, i.e., constant r_E , two extreme cases are interesting. First when r_E = r_k and the orbit is circular and the second when r_k approaches zero and the orbit approaches a straight-line segment (shown as green line in the picture). The latter case is practically impossible, but still interesting. To illustrate the effect of fixed r_E and r_k in both figures, the origin of the coordinate system is placed at the first focus.
Fig. 1 shows a family of orbits with equal energy – fixed r_E . and r_k varies from zero to 0.9 * r_E in 0.1 * r_E intervals .


Fig.1.
Fig. 2 shows a family of orbits with equal velocity momentum – fixed r_k . r_E varies from zero to 4.0 * r_k in 0.5 * r_k intervals .


Fig.2.