Astrodynamics Fundamentals. Lesson-01
Summary
TLDRThis seminar introduces the fundamentals of gravity, deriving Kepler's laws from Newton's gravitational law. It explores the universal gravitational constant and planar motion, defining velocity and acceleration in both fixed and mobile frames. The script delves into angular momentum conservation, leading to the derivation of Kepler's laws, including the constant sweep speed of the celestial body's orbital area. It further explains the elliptical orbits with the sun as a focus, using the semilatus rectum and eccentricity to describe orbital parameters. The third law's connection between a planet's orbital period and the cube of its semi-major axis is also discussed.
Takeaways
- đ The seminar introduces the fundamental concepts of gravity, the equation of motion, and Kepler's laws.
- đ Newton's law of universal gravitation is discussed, which states that the force between two masses is proportional to their product and inversely proportional to the square of the distance between them.
- đ The gravitational force is vectorial and directed along the line connecting the two bodies, with the negative sign indicating an attractive force.
- đ The universal gravitational constant (G) is given as 6.67 x 10^-11 m^3 kg^-1 s^-2.
- đ The script explains planar motion using a fixed frame (x, y, z) and a mobile frame (e1, e2, e3), defining velocity and acceleration in this context.
- đ The angular velocity (omega) is introduced as normal to the motion plane, leading to expressions for velocity and acceleration.
- đ Angular momentum (h) is shown to be constant for a system under the influence of a central force, leading to the derivation of Kepler's laws.
- đ Kepler's second law is derived, stating that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
- đ The script derives Kepler's first law, which states that the orbits of planets are elliptical with the Sun at one focus.
- đą The relationship between the semi-major axis (a), the eccentricity (e), and the semilatus rectum (p) is explored, leading to the formulation of the trajectory equation.
- âł Kepler's third law is discussed, which relates the square of the orbital period to the cube of the semi-major axis of the orbit.
Q & A
What is Newton's law of gravitation, as explained in the seminar?
-Newton's law of gravitation states that the gravitational force between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between them. The force is directed along the line connecting the bodies and is attractive.
What is the significance of the gravitational constant (G) mentioned in the lecture?
-The gravitational constant (G) is a fundamental constant in Newton's law of gravitation. It has a value of 6.67 Ă 10^-11 cubic meter per kilogram per square second and helps quantify the gravitational force between two masses.
What is the velocity of a body in planar motion, as derived in the lecture?
-The velocity in planar motion is the derivative of the position vector with respect to time. It has two components: one along the radial direction (r) and one along the angular direction (theta), which can be derived using Poisson's theorem and angular velocity.
How is the angular momentum (h) related to the motion of a celestial body?
-The angular momentum (h) is the cross product of the position vector (r) and the velocity (v). It remains constant for celestial bodies in orbital motion because the gravitational force is always directed towards the central body.
What is the purpose of the Binet formulas in the context of orbital mechanics?
-The Binet formulas are used to express velocity and acceleration in terms of geometrical parameters like r (radius) and theta (angular position) without considering time. These formulas simplify the derivation of the laws governing orbital motion.
What is Kepler's second law, and how is it derived in the lecture?
-Kepler's second law states that a celestial body covers equal areas in equal time intervals as it orbits the Sun. This law is derived using the concept of angular momentum, which remains constant throughout the orbit.
What is the 'semilatus rectum' (p) in orbital mechanics?
-The semilatus rectum (p) is a parameter that represents the radius of a celestial body's orbit when the angular position theta is pi/2. It is a key term in describing elliptical orbits.
How is the first Kepler law demonstrated in the lecture?
-The first Kepler law, which states that planetary orbits are elliptical with the Sun at one focus, is demonstrated by solving a second-order linear differential equation for the orbital radius, using Binet formulas and initial conditions.
What does the third Kepler law describe, and how is it derived?
-The third Kepler law states that the square of the orbital period is proportional to the cube of the semi-major axis of the orbit. It is derived using the relationship between angular momentum, the orbital area, and the period.
What is the general form of the differential equation used to describe orbital motion?
-The differential equation for orbital motion is a second-order linear equation for 1/r, where r is the radial distance. This equation incorporates the gravitational constant and angular momentum and leads to solutions describing elliptical orbits.
Outlines
đ Introduction to Gravitational Law and Kepler's Laws
The first lecture of the seminar introduces the fundamentals of gravity, the gravitational equation, and Kepler's laws. Newton's gravitational law is discussed, which states that the force between two masses is proportional to the product of their masses and inversely proportional to the square of the distance between them, directed along the line connecting them. The universal gravitational constant is introduced with its value. The lecture then moves to planar motion, defining velocity and acceleration in both fixed and mobile frames, and using Poisson's theorem to derive expressions for these quantities. Angular momentum is shown to be conserved in the context of Newton's law, leading to the derivation of the first binary formula, which relates angular velocity to the radius of motion squared. The lecture concludes with the derivation of the second and third binary formulas, which express the velocity component along the radius and the acceleration in terms of geometrical parameters.
đ Derivation of Binary Formulas and Kepler's Second Law
This section delves into the derivation of binary formulas, which express velocity and acceleration as functions of geometrical parameters without the need for time. Kepler's second law is then discussed, which states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. The constant rate of area swept, known as the areal velocity, is demonstrated using the first binary formula. The lecture then revisits Newton's law, eliminating the mass of the smaller body to derive an expression for acceleration. By integrating the fourth binary formula, a second-order linear differential equation is obtained, which is solved using initial conditions. The solution leads to the expression for the reciprocal of the radius as a function of the true anomaly, which is then used to derive Kepler's first law, stating that planetary orbits are elliptical with the Sun at one focus.
đ Kepler's First and Third Laws and Orbital Dynamics
The final paragraph focuses on Kepler's first law and introduces the concept of the semilatus rectum, which is a measure of the shape of the orbit. The relationship between the semilatus rectum, the major semi-axis, and the eccentricity of the orbit is explored. The trajectory equation is derived, and its implications for the radial acceleration are discussed in the context of Newton's law. The lecture then moves on to Kepler's third law, which relates the square of the orbital period to the cube of the semi-major axis. The time for a complete orbit is considered, and the area of the ellipse is used to derive a formula for the period. The constant areal velocity is used to express the period in terms of the semi-major axis and eccentricity, leading to the formulation of Kepler's third law. The lecture concludes with a summary of Kepler's contributions to understanding planetary motion.
Mindmap
Keywords
đĄGravity
đĄKepler's Laws
đĄUniversal Gravitational Constant
đĄPlanar Motion
đĄVelocity and Acceleration
đĄAngular Momentum
đĄBinet's Formulas
đĄSemilatus Rectum
đĄEccentricity
đĄOrbital Period
Highlights
Introduction to the basics of gravity, equation, and Kepler's laws.
Derivation of Kepler's laws from Newton's gravitational law.
Newton's law of universal gravitation described.
Explanation of the gravitational force being proportional to the product of masses and inversely proportional to the square of the distance.
Introduction of the universal gravitational constant (G) and its value.
Planar motion and the introduction of fixed and mobile frames for analysis.
Expression of velocity and acceleration in the mobile frame.
Derivation of the velocity vector using the Poisson theorem.
Expression of acceleration with five terms due to the radial nature of the Newtonian force.
Demonstration that angular momentum is constant in a radial force field.
Introduction of the first binary formula relating angular velocity to angular momentum.
Derivation of the second binary formula for the velocity component along theta.
Expression of the acceleration for a radial central force using binary formulas.
Introduction and explanation of the four binary formulas.
Demonstration of Kepler's second law using the concept of areal speed.
Derivation of the relationship between the gravitational force and the acceleration using the binary formulas.
Solution of the second-order linear differential equation for the radial distance.
Introduction of the semilatus rectum (p) and its significance in orbital mechanics.
Explanation of the relationship between the orbit's shape (ellipse, parabola, hyperbola) and the value of e.
Derivation of the first Kepler law stating that planets move in elliptical orbits with the sun at one focus.
Introduction of the third Kepler law relating the square of the orbital period to the cube of the semi-major axis.
Transcripts
the first lecture of the seminar is for
introducing the very basics the gravity
equation and the kepler law
actually we derive the kepler laws
from the first one
let's see how the gravitational law
discovered by newton
says
mass times acceleration is equal to the
gravity force
that force is proportional to the
product of the masses of the bodies and
inverse to the square of the distance
moreover the force is directed along the
line that connects the bodies
the negative sign
says the force is attractive
that equation is vectorial also three
scalar ones
the universal gravitational constant
capital g
is 6.67
into 10 to minus 11
cubic meter
per kilogram per square second
we refer now to a planar motion
and we introduce a fixed frame
x y z and a mobile frame e1 e2 e3
now we write the expression of the
velocity and acceleration
the velocity is the derivative of the
position vector with respect to the time
the position vector is made of a scalar
part r
and the vectorial part
r hat
that is a versus
the derivation of r hat
is the cross product between the angular
velocity omega of the mobile frame
and the velocity itself
that is due to the poisson theorem
omega
is normal to the motion plane
oriented as per e3
therefore
the derivative of r hat is
theta dot into theta hat the expression
of
the velocity
is written here
one component along e1 and one along e2
the acceleration is the derivative of
the velocity with respect to the time
and remembering that the derivative of
theta hat versus time
is omega cross theta hat
that turns into minus theta dot
into r hat
we can write
the expression of the acceleration
consisting of five terms
now we group the components along e1 and
the ones along e2
we find
the expression in red and blue
respectively
in case of newton law the acceleration
has only a component along r and the
component log theta is
zero
because the force is only radial
since the newton force is always
directed versus the central body we can
demonstrate that the angular momentum is
constant
for definition the angular momentum h
is the cross product between r and v
it refers to the unitary mass
the two derivatives on the right hand
side are both zero since the vectors are
parallel each other and since h is
constant we can calculate it at once
we find
r square
theta dot
into
k versor that means that
h is a vector perpendicular to the plane
of the motion
the first linear formula says theta dot
equals h divided r square
the component of the velocity along r
is
r dot that we can express
in that form introducing the derivative
with respect to theta we split the
derivative into two parts by plugging
the first binary formula we find this
expression on the right side that is the
second binet formula the velocity
component along theta
is r
times theta dot therefore
h over r
that is the third linear formula
the acceleration when central force is
just radial we have seen its expression
at slide number four and now we replace
the expression of theta dot from the
first binet
formula
and we introduce also the second binet
formula
for the expression of r dot
moreover we split
the derivative of r dot over time
into derivative of r dot
over theta times derivative of theta
over time that is once again the first
bini formula
altogether we get this expression in red
that is the fourth binet formula the
task of the binary formulas is to
express
velocity and acceleration as function of
the geometrical parameters only
r and theta without
the time any longer and we actually
succeeded
thanks to the mini formulas we can now
demonstrate the kepler laws once by one
let's start from the second the second
says about the ir speed the ir speed is
defined as the orbital area covered by
the celestial body
over
the time
the elementary area is defined
as one half of r
times r into theta dot
but this is actually the half of the
angular momentum
thanks to the first
binet formula
since h is constant over the orbit we
have demonstrated
that the idler speed is constant
let's come back now to the newton law
if we eliminate the mass of the small
body
and we call me the product of capital g
and capital m
we can write
that
negative me over r square
is equal to
a
the acceleration
but a
is
only radial
by plugging in the fourth binary formula
we get the second derivative
of 1 over r
with respect
to theta
plus 1 over r
is equal to me
over h square
we call
one over p that value
and now we call eta the function one
over r
and we can write a nice
second order linear differential
equation
this equation we can solve
we need just the initial conditions
and they are two
one for eta and one for eta prime
let's consider now the initial
conditions for
theta
r and for the velocity we say that for
theta
equals zero
r
is r p we are in this position
we say also that the velocity is
perpendicular to the radius
also there is no component
along r
and the component log theta is omega rp
where omega is
the local angular velocity
those conditions we need to translate
for the function eta eta 0 is 1 over r
for theta equals 0 and that is
1 over r p eta prime is for definition
the derivative of 1 over r with respect
to theta
that is
from the second binary formula
negative v r
over h
but this is zero for the initial
conditions
the general
solution of the ordinary differential
equation is eta
equals a
times
sinus of theta plus alpha plus one over
p
its derivative is
a times cosine
of theta plus alpha
the initial conditions
say eta for theta
0 must be 1 over rp and eta prime
must be 0. we have also two equations in
the two unknowns a and alpha we can get
alpha equals pi health and a equals p
minus rp
over p times r p
we can eventually write the solution
eta
function of theta as this one
remembering that
eta is 1 over r then if we reciprocate
that function
we find this expression
if we call e
the ratio between
p minus r p over p
that is a pure number then we recognize
that
this function is iconic in polar
coordinates eventually the first
kepler law is satisfied
the planet's orbits are elliptical being
the sun it's focus
what is the meaning of p
for theta equals pi half
r is equal to p
p we call the semilatus rectum
by introducing the major semi-axis a the
radius of the periapsis
is
a times e minus 1 and the radius of the
upper axis
is a
times
e plus 1.
so for theta equals
pi
that is the position of the upper axis
we get
r
equals p over
1 minus e
and if we combine the two
expressions we find the value of p being
a times one
minus e squared
let's write now the trajectory equation
we have found using p
and e if we make the inverse
and then
the second derivative over
theta we get this expression and this
expression we want to plug in
into the fourth mini formula for the
radial acceleration the result is this
expression in red
that we can compare with the value of
the acceleration
given by the newton law
and that allows to write a simple
relationship between h and p
since the orbit is a conic it can be an
ellipse a parabola or a hyperbola
assuming to deal with an ellipse
the only closed path what is the time
for a complete orbit
the area of the ellipse is
pi
times
a times b
where
b the minor semi-axis we can express as
function of a and e
and finally we get
such a formula
moreover from
the second
kepler low we know that the euler's
speed is constant and equals h
health
let's plug now the expression of h we
found in the previous slide
we get an expression
for the iolar speed
depending upon
a and e only
apart from me which is
a constant for the planet the idler
speed
is also
the total area divided by the total time
that is the period that we are looking
for
that means that the period
is
given by this expression
if we make the square
we get the third kepler law
the square of the period is proportional
to the cube of the semi-axis
of the orbit
thanks to kepler
see you next time
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