# 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.

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

### Want updates?

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

# 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.

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# 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.

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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.

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[youtube https://www.youtube.com/watch?v=MYR-0afQKo4&w=560&h=315]

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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.

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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.

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GitHub: Repository for Code Used

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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.

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[youtube https://www.youtube.com/watch?v=FwcqWdBinik&w=560&h=315]

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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.

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[youtube https://www.youtube.com/watch?v=BIYy5Ya9tgw&w=560&h=315]

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