Resource 3: Orbital Mechanics for Engineering Students

The third resource I would recommend is: 

• Title: Orbital Mechanics for Engineering Students

• Author: Howard D. Curtis

This is a cool book that's packed with information. It's often used for introductory orbital mechanics courses in undergrad. But be warned: when you open it you are going to see frightening things like quaternions, coordinate transformation matrices, Euler angles and Runge-Kutta methods. For a first reading I'd recommend the following "diet" outline, listed from the 2nd edition.

"Diet" Outline for First Reading

Chapter 1: Dynamics of point masses

Study:

• 1.1: Introduction

• 1.2: Vectors

• 1.3: Kinematics

• 1.4: Mass, force and Newton's law of gravitation

• 1.5: Newton's law of motion 

Skip:

• 1.6: Time derivatives of moving vectors

• 1.7: Relative Motion

• 1.8: Numerical integration 

Remarks: This will be a recap of what you have seen in Stewart Calculus chapter 12 and 13. Think of it as a warm up to get used to the notation of the new book. 

Chapter 2: The two-body problem

Study:

• 2.1: Introduction

• 2.2: Equations of motion in an inertial frame

• 2.3: Equations of relative motion 

• 2.4: Angular momentum and the orbit formulas

• 2.5: The energy law

• 2.6: Circular orbits (e = 0)

• 2.7: Elliptical orbits (0 < e < 1)

• 2.8: Parabolic trajectories (e= 1)

• 2.9: Hyperbolic trajectories (e > 1)

• 2.10: The perifocal frame 

Skip: 

• 2.11 The Lagrange coefficients

• 2.12 Restricted three-body problem 

Remarks: 

• This is a very long chapter, almost 60 pages. If you can get through this chapter, the rest of the "diet roadmap" is easier.

• The point of this chapter is to establish the relationship between Kepler's Laws and Newton's universal law of gravitation. From the opening paragraph: "This chapter presents the vector-based approach to the classical problem of determining the motion of two bodies due solely to their own mutual gravitation. We show that the path of one of the masses relative to the other is a conic section." 

• I suggest not getting too caught up in section 2.2 (motion of two bodies relative to a fixed frame), instead focus on 2.3  (motion of one body relative to the other body). Why: because towards the end of the chapter, we get this important statement: "Nearly all of our applications of orbital equations will be to the analysis of man-made spacecraft, all of which have a mass that is insignificant compared to the sun and planets". In other words, for most applications we are going to focus studying the motion of a small body relative to a large one; not on the motion of two bodies relative to a fixed frame. 

Chapter 3: Orbital position as a function of time

• I would skip this section on a first reading. 

• Why: the point of this chapter is to show how to calculate how long it takes to reach two different positions for a given orbit. For non-circular orbits, this is not trivial because the speed of the body varies throughout the orbit. However, from the material developed in Chapter 2, we know the total time per orbit (i.e. the period), and this is enough to get started working on some basic space flight problems. 

Chapter 4: Orbits in three dimensions 

Study:

• 4.1: Introduction

• 4.2: Geocentric right ascension-declination frame

• 4.3: State vector and the geocentric equatorial frame

• 4.4: Orbital elements and the state vector 

Skip:

• 4.5 to 4.8 

Remarks:

• A convenient aspect of the fact that the paths of orbits are conic sections is that the orbital path is constrained to a single plane; this simplifies many problems and equations. However, at the end of the day, we are still dealing with a three dimensional world, and this chapter gives techniques for how to do this.

• I would focus on 4.4 which introduces the six orbital elements of the state vector; these allow you to fully describe an orbit in three dimensions relative to the body it's orbiting, without needing Cartesian (x,y,z) coordinates. 

Chapter 5: Preliminary orbit determination

You can skip this chapter, it's about determining orbits by marking observations from the Earth. 

Chapter 6: Orbital maneuvers

Study:

• 6.1: Introduction

• 6.2: Impulse maneuvers

• 6.3: Hohmann transfer

• 6.4: Bi-elliptic Hohmann transfer

• 6.5: Phasing maneuvers

• 6.9: Plane change maneuvers

Skip:

• 6.6, 6.7, 6.8, 6.10

Remarks:

• This chapter is very important as it will give you an arsenal of maneuvers which you can sequence into a space flight 

• Regarding the skipped sections: these are useful for making more refined, fuel-efficient sequences for a flight plan, but not necessary to get started 

Chapter 11: Rocket vehicle dynamics

Study:

• 11.1: Introduction

• 11.2: Equations of motion

• 11.3: The thrust equation 

• 11.4: Rocket performance 

Remarks:

• These sections will develop the Tsiolkovsky Ideal Rocket Equation, which lets you quickly relate the amount of fuel needed to achieve a given change in velocity. This lets you calculate how much fuel is required for the various maneuvers introduced in Chapter 6. Make sure to compare the derivation to the derivation in the Braeunig book.  

Concluding remarks

If you make it through all this material, congratulations, you have entered the wonderful world of orbital mechanics, and you are in good shape to take an upper-level undergraduate or graduate-level course in this subject.

Two important topics that I excluded from this roadmap are:

• Interplanetary trajectories: this would include things like Earth-Moon transfers, Earth-Mars transfers, and planetary flybys 

• Dealing with atmospheres

At this point I would recommend practicing with your knowledge in a simulator like Orbiter or Kerbal, and then coming back to the interplanetary and atmospheric aspects of flight.