Chapter 1: Orbital Motion

Why does the gravitational force appear not to be equal and opposite between a large mass and a smaller mass?

Due to F=ma, the larger mass accelerates less due to the it having a larger mass.

In which direction does gravitational direction act?

The gravitational attraction is a central force along only the radial direction.

Why is there a negative sign in Newton’s law of gravitation?

Because the force is attractive.

How can the change in potential energy between two positions be calculated?

F[r]=-\frac{GMm}{r²} is Newton’s law of gravitation

U_f-U_i=-GMm(\frac{1}{r_f}-\frac{1}{r_i})

What is escape velocity?

For the initial speed, the escape speed v_i=v_e is defined as the minimum speed required for a smaller object to escape the gravitational field of a larger object.

What final position is chosen for calculating escape velocity and why?

r_f=\infty since at this far point, the potential energy becomes U_f=0.

What is the equation for escape velocity?

F[r]=-\frac{GMm}{r²} is Newton’s law of gravitation

v_e=\sqrt\frac{2GM}{R} where R is the distance from the centre of the larger mass to the object - this is typically the radius of the larger mass if the object is on the surface

What is the tangential speed of an object in orbit/ orbital velocity?

The speed at which the object moves along its orbital path around a central body which is crucial for maintaining the object’s stable orbit.

What is the equation for the tangential speed of an orbiting smaller mass?

F[r]=-\frac{GMm}{r²} is Newton’s law of gravitation

a_c=\frac {v²}r is the equation for centripetal acceleration

v=\sqrt\frac{GM}{r} where r is the radius of the orbit

(when inserting the equations, both acceleration and velocity act in the same direction so the negative signs cancel out)

Two objects of similar masses are orbiting a shared centre of mass? How can you find the distance between one of the objects to the centre of mass?

r_1M_1=r_2M_2 is the equation for a binary system in orbit

r_1=\frac{M_2}{M_1+M_2}R where R is the seperation between the two objects - R=r_1+r_2

Why do two objects with similar masses orbiting around a shared COM stay in line with each other?

Due to the mutual gravitational attraction of the two objects, they orbit around the COM with a shared angular frequency \omega, similar to if they were on a rigid disc. The gravitational attraction force always acts in the same line.

Where is the COM for the solar system?

For the solar system, the COM is roughly inside the sun. However, 8 large planets make this complicated as this can move the COM around.

How can the time period of two similar masses orbiting a shared COM be expressed?

F[r]=-\frac{GMm}{r²} is Newton’s law of gravitation

a_c=\frac{v²}r is the equation for centripetal acceleration

T²=(\frac{4\pi²}{G(M_1+M_2)})R³

What is 1 AU defined as?

The average distance of the Earth to the Sun

Why can we roughly ignore the other planets when calculating the orbital values of planets in our ssolar system?

The Sun’s much larger mass compared to the planets shows why we can roughly ignore the other planets when calculating the orbital values of one planet, treating the system roughly as a two-body system of the Sun and a chosen planet.

What is Kepler’s 1st law regarding motions in the solar system?

The planetary orbits are in a flat plane, and follow an ellipse with the Sun in one of the two foci.

What shape does two objects orbiting each other generally have?

Generally, the gravitational attraction between two isolated spherical objects bound to continue orbiting each other is an ellipse.

In what case do two objects orbit each other in a circle?

A circle is a special case of ellipse in which the two focal points F_1 and F_2 are both in the centre.

What are the elliptical parameters?

a = semi-major axis

b = semi-minor axis

c = the linear eccentricity

a²=b²+c²

What is eccentricity?

The eccentricity e is defined as the elongation along the semi-major axis. All ellipses have eccentricity <1.

What is Kepler’s 2nd law regarding motions in the solar system?

The area swept out by the radius vector per unit time is constant.

How does Kepter’s 2nd law come about?

Kepler’s 2nd law can be understood as a consequence of the conservation of angular momentum for an approximately isolated system of one planet and the Sun.

What is the equation for Kepler’s 2nd law?

\frac{dA}{dt}=\frac{L}{2m} where L is the constant angular momentum, m is the mass of the orbiting planet and dA is the area swpt out by the orbit in a time dt.

What is the path distance relationship for two planets which have the same area swept out by the orbit?

For areas of equal size, an area with an elliptical path closer to the Sun has a larger path distance than an area with a path further from the Sun. Thus, the orbital speed v must be higher when closer to the Sun.

What is Kepler’s 3rd law regarding motions in the solar system?

The cube of the semi-major axis a is proportional to the square of the period T.

What is an ellipse?

An orbit where the eccentricity is greater than 0. 0 is when the objects have a circular orbit. The eccentricity becomes closer to 1 as the orbit becomes more stretched out.