# Specific Energy for a Two-Body Orbit

Specific energy for a two-body orbit will be derived below. Specific energy of an orbit is constant in the two-body system. To further refine our definition, we are concerned with conservative forces only. This way specific mechanical energy is an exchange between potential and kinetic energy without drag or other perturbation losses that are non-conservative.

Specific energy was provided without proof in Eq. (1), from the previous article, "Orbital Speed for All Conic Sections," reproduced below.

 (1)

Specific energy is further reduced for all conic sections in Eq. (2) as a function of the gravitational parameter for the central body and the semimajor axis of the orbit.

 (2)

## Specific Energy Derivation

Given the two-body equation of motion, Eq. (3) we derive the specific energy for all conic orbits.

 (3)

### Step 1: Multiply by r-dot

Rearranging and multiplying by the derivative of position with respect to time,

### Step 2: Replace with derivatives of KE and PE w/r to time

As the scalar velocity multiplied by the scalar derivative of velocity term is equivalent to the derivative of kinetic energy with respect to time, we replace it as shown below on the left-hand side. The derivative of potential energy or the gravitational parameter over the radial distance is equivalent to the gravitational parameter over the radial distance squared multiplied by the scalar derivative of the position. To make the term general, we include the constant, c, which physically represents where we draw the datum for potential energy.

### Step 3: Set the reference for PE to 0 (reference level at infinity)

Setting the constant, c, to zero is equivalent to setting the datum for potential energy at infinity.

 (1)

## Function of semimajor axis and gravitational parameter

Using the periapsis and semilatus rectum's relationship to the specific angular momentum and gravitational parameter, the specific energy can be reduced to Eq. (2).

 (2)

The semimajor axis is positive for circular and elliptic orbits, infinite for a parabolic orbit, and negative for a hyperbolic orbit. Thus, the energy of each is negative, zero, and positive, respectively.

# Orbital Speed for All Conic Sections

### Specific Energy for Two-Body Orbit

Specific energy is provided, without proof, in Eq. (1) for where specific energy is a constant for any conic section. Derivation will be provided in future articles.

 (1)

Specific energy is further reduced for all conic sections in Eq. (2) as a function of the gravitational parameter for the central body and the semimajor axis of the orbit.

 (2)

### Orbital Speed for All Conic Sections

#### Elliptical and Circular

Using the specific energy equations (1) and (2), we can calculate the speed at some distance, r, from the focus as shown in Eq. (1).

 (3)

There is no variation in the distance from the focus in a circular orbit, . Reducing Eq. (3) to Eq. (4), we only need to know the circular orbit radius and gravitational parameter of the central body. The circular orbit radius is equal to the semimajor axis.

 (4)

#### Parabolic

A parabolic orbit represents the line between closed and open conic orbits. A probe given sufficient escape speed will travel on a parabolic escape trajectory from the central body. As the probe’s distance from the central body increases the speed necessary to escape decreases to zero. We determine the escape speed by comparing the specific energy of two points along this theoretical escape trajectory.

 (5)

#### Hyperbolic

Using the same trick as a parabolic orbit for the hyperbolic orbit, we must account for the excess hyperbolic speed at some infinite distance.

 (6)

# Autonomous Scheduling for Rapid Responsive Launch of Constellations

The dissertation proposal for “Autonomous Scheduling for Rapid Responsive Launch of Constellations,” by Christopher R. Simpson was successfully defended on 23 March 2020. Regular demonstrations of improvements to the model every two weeks on an agile management framework will be posted to Simpson Aerospace and Christopher R. Simpson’s doctoral committee. The proposal and addendum are available upon request.

### Abstract and Presentation

Rapid response airborne launch vehicles can provide the capability to respond to a developing situation anywhere in the world with a nanosatellite overhead in under an hour. This represents an opportunity to provide rapid response for military missions, disaster response, and rapid science return from remote/extreme physical locations. Current capabilities in the denial and tracking of space-assets limits the effectiveness of constellations already on-orbit to be agile in a military response. Constellations on-orbit can take up to a day or more for disaster data return to rescue operations personnel. Remote and rapid science return may help model Arctic cyclones which can only be accurately predicted 24 hours before they occur. To achieve time-sensitive returns from a constellation in Low Earth Orbit (LEO) scheduling algorithms for multiple near-simultaneous launches are proposed. Specifically, a mission planning system for delivery of multiple satellites from multiple similar air-launched platforms for constellation installation over any selected point optimizing for mean response time with constraints on the quality of coverage. The focus is on the scheduling of tactical fighter aircraft with airborne launch vehicles to achieve the minimum response time to fit the mission needs.

# Spring 2019 – Orbit Determination Course

Thanks to those of you that have been following the orbit determination course.

1. This course is being revamped/restarted. We will follow a typical semester schedule. We had our first lecture today.
2. I apologize to those following previously. Things got hectic with my father’s illness and qualifying exams for my Ph.D.
3. Looking forward to hearing from all of you!

Before we get started, just a few things:

• If you’re working through the material and have a question, please leave it on the lecture page on YouTube! The goal is to encourage discussion!
• I appreciate feedback! I want to make it as easy as possible for you to learn from me.

Finally,

I’ve officially passed my qualifying exams. I took my qualifiers in both Intermediate Dynamics and Space Systems. Having passed both I am officially a Ph.D. Candidate. Time to begin preparing for my thesis proposal.

Next semester will see the completion of a satellite ground station at Alabama (more on that later), the publishing of an Orbit Determination course, and my proposal.

I can’t wait to return to my research! It’s been languishing as I focused on qualifiers.