Astrodynamics Fundamentals. Lesson-01

Summary

This 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

• 📚 The seminar introduces the fundamental concepts of gravity, the equation of motion, and Kepler's laws.
• 🌌 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.
• 🔍 The gravitational force is vectorial and directed along the line connecting the two bodies, with the negative sign indicating an attractive force.
• 🌟 The universal gravitational constant (G) is given as 6.67 x 10^-11 m^3 kg^-1 s^-2.
• 📐 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.
• 🔄 The angular velocity (omega) is introduced as normal to the motion plane, leading to expressions for velocity and acceleration.
• 🌐 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.
• 📉 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.
• 📘 The script derives Kepler's first law, which states that the orbits of planets are elliptical with the Sun at one focus.
• 🔢 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.
• ⏳ 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 × 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

00:00

📚 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.

05:04

📐 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.

10:05

🌍 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.

Keywords

💡Gravity

Gravity is the natural force that attracts two objects with mass towards each other. In the context of the video, it is the fundamental concept introduced, with the gravitational law discovered by Newton stating that the force is proportional to the product of the masses of the two bodies and inversely proportional to the square of the distance between their centers. The script explains how this force is the basis for understanding planetary motion and leads to the derivation of Kepler's laws.

💡Kepler's Laws

Kepler's Laws describe the motion of planets around the sun. The video script derives these laws from the gravitational law. The first law states that planets move in elliptical orbits with the sun at one focus. The second law, or the law of equal areas, states that a line segment joining a planet and the sun sweeps out equal areas during equal intervals of time. The third law relates the time it takes for a planet to orbit the sun to the length of its orbit, specifically that the square of the orbital period is proportional to the cube of the semi-major axis.

💡Universal Gravitational Constant

The Universal Gravitational Constant, denoted by the capital letter 'G', is a physical constant that appears in the equations for gravitational force. It is approximately 6.67 x 10^-11 cubic meters per kilogram per square second. In the video, it is used to quantify the strength of the gravitational force between two masses.

💡Planar Motion

Planar motion refers to movement confined to a plane. The script introduces a fixed frame (x, y, z) and a mobile frame (e1, e2, e3) to describe the planar motion of celestial bodies. This concept is crucial for simplifying the complex three-dimensional motion of planets into a two-dimensional analysis, which is essential for deriving the equations of motion.

💡Velocity and Acceleration

Velocity is the rate of change of position with respect to time, and acceleration is the rate of change of velocity with respect to time. The script explains how to express these quantities in terms of the position vector and its derivatives. Understanding these concepts is key to analyzing the motion of objects under the influence of gravity.

💡Angular Momentum

Angular momentum is a measure of the rotational motion of an object and is conserved in the absence of external torques. The script discusses how the angular momentum is constant for a body moving under the influence of a central force, like gravity. This conservation is used to derive one of Kepler's laws, stating that a planet sweeps out equal areas in equal times.

💡Binet's Formulas

Binet's formulas are a set of equations that relate the position, velocity, and acceleration of a particle moving under the influence of a central force to the geometric parameters of its orbit. The script uses these formulas to express velocity and acceleration as functions of the orbit's shape without involving time, which is essential for demonstrating Kepler's laws.

💡Semilatus Rectum

The semilatus rectum of an ellipse is the distance from the center to the endpoint of the major axis, effectively half the length of the major axis. The script introduces this concept as a parameter in the trajectory equation of a planet's orbit, which is used to relate the shape of the orbit to the motion of the planet.

💡Eccentricity

Eccentricity is a parameter that determines the shape of an ellipse. It is a dimensionless number that ranges from 0 (for a circle) to values approaching 1 (for a very elongated ellipse). In the script, eccentricity is used to describe the shape of planetary orbits and is related to the semi-major axis and the semilatus rectum.

💡Orbital Period

The orbital period is the time taken for an object to complete one orbit around another object. The script connects the orbital period to the physical properties of the orbit, such as the semi-major axis and eccentricity, through Kepler's third law, stating that the square of the orbital period is proportional to the cube of the semi-major axis.

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