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# 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.

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# 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)
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# Executive Summary – NASA MSFC SCP/Dissertation

Executive Summary

Christopher Simpson will build a dual-use synchronized phased array utilizing a software-defined radio to test inter-formation networking and precise navigation and timing. This device will later use 24-GHz Ka-band to allow data-rates of 1-Gbps. The prototype will be presented in March 2020 at the conclusion of the effort. The payload functions as a passive radar and directed beam by utilizing electronic beam-forming, passive illumination, and network time reference protocols. During AY 2020-2021, 2 demonstration CubeSats will be built to test this game-changing technology in formation flying.

Mr. Simpson intends to collaborate with Marshall Space Flight Center (MSFC) researchers developing inter-CubeSat communication using a peer-to-peer topology. The mesh network architecture MSFC researchers are developing is intended to allow for data exchange between spacecraft with no central router. The waveform currently in use will be leveraged to reduce development risk.

### Ready to find out more?

Visit the project development page!

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# SatNOGS Station 541

### SatNOGS Station 541

"Bunny Ears," one of the SatNOGS ground stations I run has been re-established. The ground station consists of a Raspberry Pi 3, RTL-SDR, and a TV dipole antenna, "bunny ears."

### 541 - Bunny Ears

Check out any of the observations currently scheduled!

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# Marshall Space Flight Center Fiscal Year 2019 Student Collaboration Projects (SCPs) – SIMPSON

Visit the project page for the most up-to-date information!

NASA Marshall Space Flight Center (MSFC) released a call on June 3 for proposals to collaborate with promising students and leverage ongoing work to explore new, innovative applications of that ongoing work.

Executive Summary

Christopher Simpson will build a 130-430 MHz dual-use software-defined radio to test inter-formation networking and precise navigation and timing. This device will later use 24-GHz Ka-band to allow data-rates of 1-Gbps. The prototype will be presented in March 2020 at the conclusion of the effort. The payload functions as a passive radar and directed beam by utilizing electronic beam-forming, passive illumination, and network time reference protocols. During AY 2019-2020, 2 demonstration nodes will be built to test this game-changing technology in formation flying.

Mr. Simpson intends to collaborate with Marshall Space Flight Center researchers developing inter-CubeSat communication using a peer-to-peer topology. The mesh network architecture MSFC researchers are developing is intended to allow for data exchange between spacecraft with no central router. The waveform currently in use will be leveraged to reduce development risk.

This proposal addresses NASA Roadmap 2015 - TA 5.5.1.1, Intelligent Multipurpose Software Defined Radio and enhances a return to the lunar surface by addressing LEAG - Strategic Knowledge Gap (SKG) Theme 1-D Polar Resources 7.

Addressing the Scientific and Technical Challenges

1.Track and communicate with other nodes (Satellites in the formation)

• Simulate on ground the tracking and communication capability this network will provide

2.Expanding Network Time Reference (NRT) to communication systems to reduce reliance on external time references and improve navigation.

• Use this NRT to electronically form the beam and transmit/track satellite.
• Use same antenna for communication/radar.

3.Reduce required SWaP while improving technical merit.

• Electronic beam-steering for inter-formation tracking and communication networking has not been demonstrated previously, see NASA Small Satellite Database.
• Missions are in the work to demonstrate inter-formation network

Budget and Time Constraints

\$6,000 for materials (adjusted for risk/price increase)
Table in presentation.
20 Weeks (10 Sprints/625 hrs)

I intend to utilize Scrum planning to utilize an AGILE development. I will finish January 12 if everything occurs ideally. This leaves me with an extra 9 weeks of overage or another 281.25 hours of development.

Documents:

Synchronized Phased Array Software Defined Radio

NASA-SCP-Response_SIMPSON

NASA-SCP-Response_Storyboard_Outline

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# Learning Creo Parametric (New CAD tools!)

I recently installed and started teaching myself the Creo suite of tools. I needed a replacement for AutoDesk Inventor. I’ve posted the finished product of the tutorials for building a piston/ piston shaft. I would like to reach the same capability I previously held with Inventor. For those of you not familiar with Creo, Wikipedia offers this:

Creo Elements/Pro and Creo Parametric compete directly with CATIA, Siemens NX/Solidedge, and SolidWorks. The Creo suite of apps replace and supersede PTC’s products formerly known as Pro/ENGINEER, CoCreate, and ProductView.

I previously used AutoDesk Inventor to make the Gulfstream GV/ G550 model (Gulfstream G-V CAD). Dr. Charles O’Neill has reproduced a version of this model in CATIA. The article describing the model is here: https://charles-oneill.com/blog/gulfstream-gv-g550-cad-model/ His model is available on GrabCad: https://grabcad.com/library/gulfstream-gv-g550-low-fidelity-2

Gulfstream GV / G550 CAD Model

Engineers/pilots will notice, on my model, the abscence of wingtips and the exact airfoil is reproduced as best as possible for being lofted from drawings. This drawing was intended as low fidelity to facilitate a proposal. It meets those requirements.

Completing the Creo tutorial required some breakdown between both the text and the videos provided. Completed exercises are shown below.

A piston created in Creo (Creo Beginner Exercise 1)

A crankshaft to emphasis patterns and simplifying (Creo Beginner Exercise 2)

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# Spring 2019/Lecture 12/Real Measurements 2 – 22 Feb 2019

Two way ranging and Doppler systems are summarized. Differenced measurements or “differencing,” is explained. For additional explanation on differencing see Penn State’s course for Geospatial and GNSS professionals (https://www.e-education.psu.edu/geog862/node/1727).

There was some difficulty with projecting the slides to the screen so they have been added after the lecture was recorded.

Previous lectures:

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# Spring 2019/Lecture 10/Conceptual Example – 18 Feb 2019

Return from recording issues. An example illustrating the previous discussions on real-world limitations of observations is examined.

Previous lectures:

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# Spring 2019/Lecture 9/Conceptual Measurements – 15 Feb 2019

Return from recording issues. Real-world limitations on ideal observations are discussed. An example illustrating these discussions is prepared for the next lecture.

Previous lectures:

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# Spring 2019/Homework 2 Solution

Demonstrate understanding of orbital mechanics necessary to complete orbit determination course. In problem 1, position and velocity are converted between osculating elements and sub-satellite points. In problem 2, the receiver measurements confirm the node location varies over time. In problem 3 the equations of motion are numerically integrated for a GLONASS satellite for one day.

GitHub: Repository for Code Used

Previous lectures:

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# Spring 2019/Lecture 8/Simulating Ideal Measurements – 13 Feb 2019

Continuing from ideal range and range rate measurements we examine how this applies in the larger context of orbit determination. We use examples to demonstrate real-world application. My apologies again for the difficulties I had bringing this recording to you.

Previous lectures:

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# Spring 2019/Lecture 7/Ideal and Conceptual Measurements – 11 Feb 2019

What is an ideal measurement? Specifically what is an ideal range and/or range rate measurement? What’s the difference between observed and computed measurements? Why is it important? My apologies again for the difficulties I had bringing this recording to you.

Previous lectures:

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# Spring 2019/Lecture 6/Coordinate Systems and Time – 8 Feb 2019

You should have turned in your assignment by this lecture. My apologies for the issues getting the recordings online. We covered different Earth-bound reference frames and timing systems.

Previous lectures:

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# Spring 2019/Technical Difficulties

I’m having some technical difficulties with recording the lecture. I’ll re-record all three and post them for you later. For now, enjoy lecture 5.

Also, you should have finished your homework and turned it in. Solutions will be posted soon.

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# Spring 2019/Lecture 5/Perturbed Motion – 6 Feb 2019

Sorry for the missed lectures on Friday and Monday. I was out sick. Assignments are due on Friday. We covered perturbations to orbital motion. We examined contributions from gravitational and nongravitational sources to the two-body motion.

Slides: L5 Slides – Perturbed Motion

Previous lectures:

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# Spring 2019/Lecture 4/Two Body Problem – 30 Jan 2019

We resumed today with orbital mechanics. We covered the two-body problem, introduced Kepler’s problem (time doesn’t relate well to true anomaly), and sprinted to the state transition matrix. We will resume with perturbations and additional bodies considered on Friday.

Slides: L4 Slides – Two Body Problem

Previous lectures:

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# Spring 2019/Lecture 1/Orbit Determination Concepts – 23 Jan 2019

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.

Lecture Slides:

Orbit Determination Concepts – Lecture 1

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# 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,

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I attended the 2018 International Geoscience And Remote Sensing Symposium (IGARSS 2018) in July; presenting on the recently proposed 50 CubeSat constellation to sound the Antarctic ice sheets. There are still large gaps in ice thickness data despite more than 50 years of airborne radar sounding. A satellite mission presents an opportunity to gain complete coverage of the ice sheets. Some key features of the constellation include a 50 m and 1 m along-track and cross-track separation, respectively, a Ka-band radar and downlink device, and a 150 MHz sounder.

Image credit: (2018/Charles O’Neill)

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# Lecture 3 – Orbital Mechanics Review B

[contact-form][contact-field label=”Name” type=”name” required=”1″ /][contact-field label=”Email” type=”email” required=”1″ /][/contact-form]

### Lecture

I pick up again by reviewing the solution to the problem assigned during Lecture 1. (The link will take you to a solution using C++ on GitHub). A few common coordinate systems and reference frames are introduced, orbital perturbations are introduced, and an example problem to be solved in Lecture 4 is given to the class to start on.

Lecture 2

Lecture 1

### Resources

Lecture 3 – Review Of Orbital Mechanics B

Lecture 2 – Orbital Mechanics Review A

Lecture 1 – Orbit Determination Concepts (slides)

AppendixA-ProbabilityAndStatistics

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# OD – HW 1 Solution

[contact-form][contact-field label=”Name” type=”name” required=”1″ /][contact-field label=”Email” type=”email” required=”1″ /][/contact-form]

## Solution to Homework 1

Problem 1 provides the solution for us. We are learning how to use iterative methods to estimate the state vector. In this case we will use the Newton-Raphson root-finding method to solve for the problem. An Excel sheet is provided that walks through the first three iterations as an illustration. C++ code is provided on GitHub that will solve for the final solution and show the number of iterations.

Homework Solution 1

Excel – Visual Iterative Solution

C++ Solution

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# OD – Postponement/Ans.

I’ve been traveling and haven’t been able to finish the last part of Orbital Mechanics Review. Feel free to review Dr. Russell’s work, prior to Monday. Full solution will be posted Monday as well. If you’re answers don’t match up; start a discussion in the comments below.

 X0 1 Y0 8 Xdot 2 Ydot 1 g 0.5 Xs 1 Ys 1
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# Lecture 2 – Orbital Mechanics Review A

[contact-form][contact-field label=”Email” type=”email” required=”1″ /][/contact-form]

### Lecture

We review orbital mechanics and Newton’s law of gravitation to prepare for orbit determination. We will cover the two body problem, orbital elements, and perturbing accelerations. We won’t finish the entire lecture today. We will continue on Monday.

Lecture 1

### Resources

Lecture 2 – Orbital Mechanics Review A

Lecture 1 – Orbit Determination Concepts (slides)

AppendixA-ProbabilityAndStatistics

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# Lecture 1 – Orbit Determination Concepts

[contact-form][contact-field label=”Email” type=”email” required=”1″ /][/contact-form]

### 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

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# Statistical Orbit Determination

[contact-form][contact-field label=”Email” type=”email” required=”1″ /][/contact-form]

### 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

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# Space Operations and Support Technical Committee

Today I received official notice: I am a member of the AIAA Space Operations and Support Technical Committee (SOSTC). The SOSTC Charter:

The Space Operations and Support Technical Committee (SOSTC) is concerned with all aspects of civil, military, and commercial space operations and support, including direct and supporting operations, the systems and software affecting operations, and space operations and operational risk management. The SOSTC addresses all types of space operations, including manned and unmanned space operations from low Earth-orbiting to deep-space systems. It is involved with all phases of mission operations, including pre-launch and launch activities, early mission commissioning activities, on-orbit activities, cruise and encounter activities, post-landing activities, and end-of-life operations. The SOSTC likewise addresses space related operational support activities, including training, servicing, mission planning, flight dynamics, telemetry transmission, command and control, and data handling, processing, analysis, and storage.

I’m very thankful for this opportunity.

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# Post-Flight Analysis Report (PFAR) of RX1

SUMMARY:

Christopher R. Simpson built a rocket to pass his Level 1 (L1) certification from the National Association of Rocketry (NAR). The rocket was a kit from Madcow Rocketry; the “Frenzy,” [1]. The RX1 used an Aerotech H550ST-14A, “Super Thunder,” motor with a total impulse of 71.9 lb-sec and a burn time of 0.57 sec. Construction of the rocket, flight, and recovery are reviewed to analyze and critique operations.

Post-Flight Analysis Report (PFAR) attached here: PFAR-RX1 (26 Feb 2018)

ACKNOWLEDGEMENTS:

A big thank you to Karson Holmes for certifying/critiquing me and William Ledbetter for making the trip to watch the fun take off! Also, a special thanks to Alabama Rocketry for allowing me to use their adapter.

RESOURCES:

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# FreeFlyer Demonstration: 2:00pm Sep. 28, 2017 for UA faculty and students

I will be giving a demonstration of FreeFlyer on Thursday, September 28 at 2:00 pm in SERC 3070. Faculty and Students feel free to drop on by! I have attached the flyer, here: FreeFlyerWithAttitude

Christopher Simpson will present a FreeFlyer demonstration, “FreeFlyer with Attitude,” on Thursday, September 28, 2017 at 2:00 pm in SERC 3070. FreeFlyer with Attitude will showcase the high-fidelity flight dynamics software with a Earth imaging satellite mission plan with specific pointing requirements. FreeFlyer is currently used on several NASA missions, including the Magnetosphere Multiscale (MMS) mission which set the record for closest flying formation at 7.2 km in September of 2016. Mr. Simpson recently interned with a.i. solutions, Inc. over the summer and worked with the FreeFlyer Tech Support team. He recently graduated with his B.S. in Aerospace Engineering and Mechanics from The University of Alabama in May 2016. He was recently awarded a SMART scholarship from the Naval Air Warfare Center – Weapons Division, China Lake. He is pursuing his Ph.D. at The University of Alabama under Dr. Charles O’Neill.

 WHAT WHO WHERE WHEN Showcase of FreeFlyer Students & Faculty SERC 3070 2:00 PM September 28, 2017 A high fidelity flight dynamics software comparable to STK used on multiple NASA missions, including the ISS, MMS, OSIRIS-Rex, and for the SLS All with an interest in spacecraft and astrodynamics are welcome

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# 2017 ESPRMC Graduate Research Symposium – The University of Alabama

I will be presenting “Benefits of Tracking Aids on a 1U CubeSat,” on Thursday, April 13 at the 2017 ESPRMC Graduate Research Symposium. Dr. O’Neill was my co-author. I hope to see you there.

Abstract:

Incoporating active/passive tracking aids into the design of a university/high school CubeSat mission promotes good space stewardship. Tracking aids are necessary for improved tracking covariance of CubeSats. Tracking aid support and design-space cost are covered. Reflectarrays, patch array(s), and deployable antennas show the potential benefit of transmitting data over S-band frequencies and tracking aids that enhance the mission capabilities. Passive and active tracking aids with low impact on the mission provide reduced covariance of CubeSats orbit tracks shown through use of modeling tools.

# 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/Lecture 11/Conceptual Example – 20 Feb 2019

The environment and relativity effects on radio and optical communications are introduced.  One-way range measurement systems are introduced. GPS is provided as an example but it still applies to GLONASS and Galileo. Two-way range, Doppler, and differenced measurements are considered next.