# 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)

# Lecture 1 – Orbit Determination Concepts

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### Lecture

The inherent characteristics of an orbit determination (OD) problem are introduced. Dynamic state estimation, observations, linearization, and the state transition matrix are discussed. At the end, I have left a practice problem that we will review on Friday, 8 June. We throw a satellite up and watch it come down while introducing some important concepts.

### Resources

Lecture 1 – Orbit Determination Concepts (slides)

AppendixA-ProbabilityAndStatistics

# Statistical Orbit Determination

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### Class time/Preliminary Notes

I will be teaching a statistical orbit determination course this summer. This will be on my own time. All lectures will be posted to YouTube. I will be teaching the course out of Bob Schutz’s, Byron Tapley’s, and George H. Born’s, Statistical Orbit Determination. Feel free to use any textbook you desire but the problems and solutions will be assigned from this text. I have included some precursor notes in question and answer format on statistics and probability below.

AppendixA-ProbabilityAndStatistics

### Syllabus

AEM_StatisticalOrbitDetermination_Syllabus_CRS

STATISTICAL ORBIT DETERMINATION

EXECUTIVE SUMMARY:

Orbit Determination (OD) is the problem of determining the best estimate of the state of a spacecraft whose initial state is unknown, from observations influenced by random and systematic errors, using a mathematical model that is not exact. Mordern OD is used to support all space missions from JSpOC’s observations of artificial Earth satellites to the International Space Station’s trajectory planning incorporating elements of probability, statistics, and matrix theory. A special projects class is needed to cover this vital part of the space curriculum that arguably makes the backbone of any space program.

DISCUSSION:

Modern OD approaches have been developed by the NASA Jet Propulsion Laboratory (JPL) to fulfill Earth and planetary navigation requirements and at the NASA Goddard Space Flight Center (GSFC) and the Department of Defense Naval Surface Weapons Center in applications of satellite tracking to problems in geodesy, geodynamics, and oceanography. The Joint Space Operations Center (JSpOC) at Vandenberg Air Force Base, the Conjunction Assessment Risk Analysis (CARA) at GSFC, and Trajectory Operation Officers (TOPO) at Johnson Space Center (JSC) use modern OD techniques in applications of satellite tracking, conjunction assessment, and protecting vital assets from the International Space Station to the National Reconnaissance Office (NRO) spy satellites.

Clearly, OD is an important part of any space mission. The proposed class will use the classical text, Statistical Orbit Determination, by Drs. Byron Tapley, Bob Schutz, and George Born. This basic OD course will cover:

• Introduction to OD problem
• Dynamic system and associated state
• Observations are non-linear functions of state variables
• Classical well-determined approach
• Modern over-determined approach
• Observations to measure satellite motion
• Ground-based systems: laser, radiometric, etc.
• Space-based systems: GPS, etc.
• Error sources and media corrections
• Non-linear OD reduced to linear state estimation
• Application of linear system theory
• Incorporation of algorithms to computational environment
• Sequential processing of observations
• Control of real-time processes

This will be supported by background and supplemental information in:

• Probability and Statistics
• Review of Matrix Concepts
• Examples of State Noise and Dynamic Model Compensation
• Solution of the Linearized Equations of Motion

Students can expect to incorporate their classroom knowledge into real-life by building optical and radiometric sensors supporting The University of Alabama’s new satellite ground station.

LECTURES:

Lecture 1 – Orbit Determination Concepts

Lecture 2 – Orbital Mechanics Review

# Coming Soon: Orbital Mechanics/Astrodynamic Problem Solutions

While in the midst of preparing for a journal paper I decided that I wanted to showcase my abilities. I will solve all the problems from Vladimir Chobotov’s Orbital Mechanics, Third Edition, and Richard Battin’s An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition and post the solutions online. I hope to have this done by January 2.

Not only will this be a good review for myself but it will showcase my abilities to solve problems relating to the field I want to enter. Hopefully, it will prove to be a valuable tool in the future.

As for the featured picture: I am in the process of getting myself certified Level 1 with STK. I dropped this scenario into Cesium while I was practicing and exploring STK before the exam. My exam is due December 22. I will let you know the results soon!

# CODER 2016: Panelist/Presentation/Experience

I presented Tuesday at the Center for Orbital Debris Education and Research on “CubeSat Network for Prediction and Tracking of Orbital Debris.” The key points:

1. Conceptual Stage of Development
2. LIDAR ~700W and CubeSat ~200 W
3. Assuming UV laser, 355 nm, resolution of 1 mm would require a 2 m telescope diameter for a range of 5 km

Presentation is attached here: simpson_v2

Several topics that are of interest specifically to my presentation:

• Research that suggests that the satellite operators that do not follow good practices will be the primary source of new collisions in the future. Roger Thompson, The Aerospace Corporation, spoke to this on my panel.
• Space is about to get a lot more crowded. To put it in context their have been 56 launches this year. SpaceX filed for 4400 satellites with the FCC Wednesday.
• There is a strong need for knowledge of conjunctions and good covariance analysis. I.E. there seems to be the need for secondary analysis for conjunction warnings from JSpOC. On my panel Don Greiman, Schafer Commercial Space Situational Awareness Team, spoke to this from the commercial side and Ryan Shepperd as analysis in-house for Iridium.

Talk about a huge learning experience. I hope to take what I’ve learned back to The University of Alabama and disseminate this knowledge among our academic community. I also plan to stay in touch with the contacts I’ve made to help expand my horizons.

# Presenter for CODER 2016 Workshop

CODER is the Center for Orbital Debris for Education and Research. CODER is having a Workshop and accepted our, Dr. O’Neill and I, abstract for CubeSat Network for Orbital Debris Tracking and Prediction. I will be a presenter for the Space Situational Awareness Session. Here’s the link for my session and bio: CODER 2016 Sessions

The conference is hosted at the University of Maryland on November 15-17. If you’re headed to the conference hope to see you there!