Introduction to Orbital Mechanics and Spacecraft Attitudes for Thermal Engineers Presented by: Steven L. Rickman NASA Technical Fellow for Passive Thermal NASA Engineering and Safety Center 18-20 August 2020 1 Video Credit: DSCOVR: EPIC Team Differences in Lesson Scope Introduction to Orbital Mechanics and Spacecraft Attitudes for Thermal Engineers • 2-Body Problem Introduction to On- • Conservation of Energy • Conservation of Orbit Thermal Momentum Environments • Kepler’s Laws • Sun Synchronous Orbit • Beta • Geostationary Orbit Angle • Molniya Orbit • Eclipse • Advanced Orbit Concepts • Solar Flux • Albedo • Reference Frames • Simple • OLR • Spacecraft Attitude View Transformations Factor • General View Factor Equation 2 References and Credits References and image sources are cited on each slide. Unless otherwise credited, animations were developed by the author using Copernicus 4.6. Microsoft® Clip Art was used in the presentation. 3 Disclaimers The equations and physical constants used in this lesson are for instructional purposes only and may involve simplifications and/or approximations and should not be used for design, analysis or mission planning purposes. While screen shots or analyses resulting from analysis tools are presented, no endorsement for any tool is made. 4 Introduction Orbiting spacecraft are subject to a variety of environments. Knowledge of the orbit is required to quantify the solar, albedo and planetary (also called outgoing longwave radiation, or OLR) fluxes. Some specific questions that might arise are: • How close (or how far) does the planet/spacecraft pass from the sun? • How close (or how far) does a spacecraft pass from a planet and how does it affect orbital heating to spacecraft surfaces? • How long does the spacecraft spend in eclipse during each orbit? • At what angle does the solar flux impinge on the orbit plane (훽 angle) and how does that affect the thermal environment? • What path does a spacecraft take between planets and how does the solar flux change during that transfer? • Why is one type of orbit used for some spacecraft and another type used for others (e.g., sun synchronous versus geostationary)? • What factors can make an orbit change over time and how might that affect the thermal environment? • What type of thermal environment extremes will the spacecraft experience? 5 Introduction Orbit information alone is insufficient to determine how the environment affects the spacecraft. Spacecraft orientation (or “attitude”) and orbit information is required to determine which spacecraft surfaces experience a given thermal environment. Spacecraft attitude and orbit information are required to determine the view factor to the central body which is required for planetary and albedo flux calculations to a spacecraft surface. What are the effects on the heating fluxes experienced by a spacecraft due to the attitude reference frame (e.g., celestial inertial versus local vertical – local horizontal reference frames)? What spacecraft orientation(s) provide favorable thermal conditions for spacecraft components? Orbits and spacecraft attitudes must be considered together for a successful spacecraft and mission design. 6 Scope of this Lesson Orbits Spacecraft attitudes Governing differential equation Conservation of specific mechanical energy Conservation of specific angular momentum Kepler’s laws Perturbations Consequences for the thermal environment. 7 Lesson Contents (1 of 5) Part 1 -- Review of Scalar, Vector and Matrix Operations Scalars and Vectors Cartesian Coordinates Vector Dot Product Vector Cross Product Unit Vectors Coordinate Transformations The Euler Angle Sequence Rotation Sequences Forming the Transformation Matrix Stacking the Transformations 8 Lesson Contents (2 of 5) Part 2 -- The Two Body Problem Aside: Anatomy of an Orbit Aside: History Strategy Newton’s Laws Relative Motion Aside: Some Useful Expansions of Terms Conservation of Specific Mechanical Energy Conservation of Specific Angular Momentum Kepler’s First Law Circular Orbit Elliptical Orbit Parabolic Orbit Hyperbolic Orbit Example: Determining Solar Flux Using Kepler’s First Law Kepler’s Second Law Example: Using Kepler’s Second Law to Determine How Solar Flux Varies with Time Kepler’s Third Law Example: Determine Planet Orbital Periods Using Kepler’s Third Law Example: Geostationary Orbit 9 Lesson Contents (3 of 5) Part 3 -- Perturbed Orbits Governing Differential Equation Perturbations Precession of the Ascending Node Example: Sun Synchronous Orbit Precession of the Periapsis Example: Molniya Orbit The Effect of Orbit Perturbations on the Thermal Environment The Beta Angle Calculating the Beta Angle Variation of the Beta Angle Due to Seasonal Variation and Orbit Precession Consequences of Beta Angle Variation Eclipse Calculating Umbral Eclipse Entry and Exit Angles Fraction of Orbit Spent in Sunlight/Eclipse Example: Eclipse Season for a Geostationary Orbit Example: ISS Orbit Example: Sun Synchronous Orbit 10 Lesson Contents (4 of 5) Part 4 -- Advanced Orbit Concepts Transfer Orbit Orbit Plane Change Aerobraking Orbit Gravity Assists The Restricted Three-Body Problem Halo Orbits Artemis I Gateway (Near Rectilinear Halo Orbit) 11 Lesson Contents (5 of 5) Part 5 -- Spacecraft Attitudes Reference Frames Vehicle Body Axes Local Vertical-Local Horizontal (LVLH) Celestial Inertial (CI) Comparing LVLH and CI Reference Frames Attitude Transformation Strategy Transforming Attitudes in CI Transforming Attitudes from LVLH into CI Aside: View Factor to Planet as a Function of Orbit and Attitude Example: Heating to Spacecraft Surfaces as a Function of Orbit and Attitude Conclusion Acknowledgements To Contact the Author 12 Part 1 -- Review of Scalar, Vector, and Matrix Operations 13 Part 1 -- Content Part 1 of this lesson is a review of mathematical operations we will need in our study of orbital mechanics and spacecraft attitudes. We will begin with a review of scalars and vectors. After a brief review of Cartesian and Polar coordinates, we’ll consider vector dot and cross products, units vectors, coordinate transformations with particular focus on the Euler angle sequence, forming transformation matrices and, finally, stacking transformations. 14 Scalars and Vectors A scalar has a magnitude whereas a vector has, both, a magnitude and a direction. As an example, speed is a scalar and has a magnitude (e.g., 30 m/s) but velocity is a vector and has a magnitude and direction (e.g., 30 m/s in the 푥-direction). We will use, both, scalars and vectors in our study of orbital mechanics and attitudes. 15 Cartesian Coordinates Consider the Cartesian coordinate system. 풌෡ Each axis is orthogonal to the others. 푥, 푦, 푧 Any point in the coordinate system may be 푧 described by three coordinates 푥, 푦, 푧 . 풋Ƹ 푥 To aid in describing the amount of travel in 푦 each orthogonal direction, we specify unit 풊Ƹ vectors (풊Ƹ, 풋,Ƹ 풌෡). 16 Polar Coordinates Polar coordinates specify 90° the location of a point using two coordinates, a distance from the origin, 푟 and an 푟 angle, 휃. 휃 180° 0° Polar coordinates will be especially useful in our discussion of orbits. 270° 17 Vectors The vector, 풓ത can be expressed in Cartesian 풌෡ coordinates as: 풓ത = 푥풊Ƹ + 푦풋Ƹ + 푧풌෡ 푥, 푦, 푧 풓ത 푧 풋Ƹ The magnitude of the vector, 풓ത is given by: 푥 푦 풊Ƹ 풓ത = 푥2 + 푦2 + 푧2 18 Useful Vector Operations Consider the two vectors shown at the right… 풌෡ ෡ 풓ത = 푥푟 − 0 풊Ƹ + 푦푟 − 0 풋Ƹ + 푧푟 − 0 풌 푥 , 푦 , 푧 푥푟, 푦푟, 푧푟 ෡ 푠 푠 푠 = 푥푟풊Ƹ + 푦푟풋Ƹ + 푧푟풌 풔ത 휃 풓ത 풋Ƹ 0,0,0 ෡ 풔ത = 푥푠 − 0 풊Ƹ + 푦푠 − 0 풋Ƹ + 푧푠 − 0 풌 ෡ 풊Ƹ = 푥푠풊Ƹ + 푦푠풋Ƹ + 푧푠풌 19 Vector Dot Product The dot product of two vectors, 풓ത and 풔ത, is a scalar given by… 풌෡ 푥 , 푦 , 푧 풓ത ∙ 풔ത = 풓ത 풔ത cos 휃 푥푠, 푦푠, 푧푠 푟 푟 푟 풔ത 휃 풓ത 풋Ƹ For the vectors shown at the right… 0,0,0 풓ത ∙ 풔ത = 푥푟푥푠 + 푦푟푦푠 + 푧푟푧푠 풊Ƹ 20 Vector Cross Product The cross product of two vectors, 풓ത and 풔ത, is a vector given by… 풌෡ 푥 , 푦 , 푧 풊Ƹ 풋Ƹ 풌෡ 푥푠, 푦푠, 푧푠 푟 푟 푟 풓ത × 풔ത = 푥 푦 푧 풔ത 푟 푟 푟 휃 풓ത 푥푠 푦푠 푧푠 풋Ƹ 0,0,0 For the vectors shown at the right… 풊Ƹ ෡ 풓ത × 풔ത = 푦푟푧푠 − 푧푟푦푠 풊Ƹ − 푥푟푧푠 − 푧푟푥푠 풋Ƹ + 푥푟푦푠 − 푦푟푥푠 풌 21 Unit Vectors As the name implies, a unit vector is a 푥푟, 푦푟, 푧푟 vector with one unit of length; 풌෡ 풓ത To form a unit vector, 풓ො in the direction of 풓ො 풓ത… 푥푠, 푦푠, 푧푠 풓ത 풓ത 풋Ƹ 풓ො = = 풓ത 2 2 2 푥푟 − 푥푠 + 푦푟 − 푦푠 + 푧푟 − 푧푠 풊Ƹ 22 Coordinate Transformations Analysis of spacecraft in orbit in a specified attitude requires an understanding of coordinate system transformations. The position in orbit and the position with respect to heating sources and the eclipse is determined using coordinate system transformations. Additional transformations are performed to orient the spacecraft as desired at any given point in orbit. These transformations are performed as Euler angle sequences. 23 The Euler Angle Sequence An Euler angle sequence is a sequence of rotations of a rigid body with respect to a fixed coordinate system. The sequence is order dependent – that is, changing the order of the rotations will affect the resulting transformation. We will rely on Euler angle transformations considerably during this lesson. They are easily executed using multiplication of 3 × 3 matrices. Some info from: https://en.wikipedia.org/wiki/Euler_angles 24 Rotation Sequences However, we need to be specific about the type of rotation we seek – there are two possibilities: Rotation of the axes, or Rotation of an object relative to fixed axes. We ultimately seek a rotation of an object relative to fixed axes. From: mathworld.wolfram.com/RotationMatrix.html 25 Rotation Sequences Consider the vector 푷ഥ which is at an 푦 angle, 휙 from the 푥-axis in the fixed 푷ഥ′ 푥′, 푦′ coordinate system. We wish to transform this vector into 푟 푷ഥ′ by rotating it through angle, 휃 in 푷ഥ (푥, 푦) the same fixed coordinate system.