Artemis Luna 101 By: Kushal Shah, Jesus Ramos, Viral Patel,

Omar Franco, Mark Saleh, Carlos Solórzano

3/15/2013

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Artemis Luna 101 Mission Objective: To develop a spaceflight mission concept to use solar reflective mirrors in orbit around the Moon to provide a constant illumination power of 200 kW on the South Pole of the Moon using Sun’s Energy.

I. Mission Overview(Omar & Mark & Kushal) The Artemis Luna Mission is a mission concept, where four satellite having large mirrors attached to it

will orbit the moon in order to focus the sun’s rays onto the moon’s South Pole and to produce 200 kW of power while keeping costs minimal during all the time. This satellite will orbit around the moon in the elliptical orbit with semi major axis of 5439.94 km, eccentricity of 0.67 in the inertial frame orientation of J2000 attached to the center of the moon. In case of a satellite malfunction or failure, 76% of the targeted illumination power (200 kW) will still be focused upon the South Pole. To control the attitude of the satellites in the orbit, momentum wheels, aka reaction wheels, will be used, and to measure the rotation of inertial frame gyroscopes and MEM devices will be used. These momentum wheels orient the principle body frame attached to satellite about third axis to achieve needed orientation of the satellite to achieve the mission goal by transferring momentum and following newton’s third law of action-reaction law. The initial rocket, Atlas V-431, from earth to low earth orbit will be launched on 26th March 2013 at 15:04 (CST) from the Kennedy Space Center in the Low earth orbit of 1000 km. The site was chosen as it was convenient for the orbital transfer since it aligned with the orbital plane of the moon. During the first period in LEO, all the satellites will perform phasing maneuvers to achieve initial desired positions derived from the required positions at the beginning of the Hohmann transfer in the later stage. From LEO, satellites will perform Hohmann transfer and plane change maneuver, once in the orbit designed around the moon, to attain designed positions in the orbit around the moon to achieve the mission objective. The total ΔV for the mission is 15.94 km/s and total mass of the payload for Atlas V-431 will be 12,201 kg, including 500kg structure mass of each satellite and 100kg of payload mass of each satellite.

In order for the mission to be successful, several things were determined once the satellites reached the moon: the number of satellites, their configuration, orientations, and the orbits they take around the moon. The first step for the mission modeling was to determine the orientation and the locations of the satellites in the orbit around the moon. Second task of the mission was to find targeted positions for the space-shuttle that is achievable by a Hohmann transfer. While modeling this section, the alignment of the space-shuttle and the moon’s position at intermediate launch and the final arrival time were considered and the data table of launch time and arrival time was created. The next step was choosing our initial launch date and position that the first Hohmann orbit will occur based upon designed locations obtained from the first step and the desired orbital phasing in the LEO orbit. From the calculations of the desired positions, maneuvers were modeled and the amount of and total ΔV and total fuel, used in the process from launch to the targeted orbit, were calculated. Lastly, based on the amount of the payload the type of rocket was chosen.

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Table of Contents Mission Objective .................................................................................................................................. 1

I. Mission Overview(Omar & Mark).............................................................................................. 1

II. Introduction(Viral) ..................................................................................................................... 6

III. Modeling Assumptions (Carlos) ............................................................................................... 8

1. For The Motion Of The Moon (Carlos)............................................................................................... 8

2. For The Modeling (Jesus) .................................................................................................................... 8

3. Parameters for Earth & Moon (Mark) ................................................................................................. 8

IV. Launch of S/C from Earth to Moon (Technical Details: Kushal Shah & Viral)........... 9

1. Introduction: (Kushal) ......................................................................................................................... 9

2. Earth Centric Inertial (ECI) Frame and Coordinate System Description (Author: Kushal & Viral) .. 9

3. Selecting Launch Site (Author: Kushal ) ........................................................................................... 10

4. Sending the Spacecraft to its First Orbit Around The Earth (Author: Kushal) ................................. 10

5. Phasing Maneuvers (Author: Omar & Mark) .................................................................................... 12

6. Describing Moon’s Orbit (Mark & Omar) ........................................................................................ 13

7. The Hohmann Transfer ( Kushal ) ..................................................................................................... 13

7.1 The Hohmann Transfer Concept (Author: Viral):...................................................................... 13

7.2 Hohmann Transfer Positions and Timing: (Mark, Kushal & Omar) ......................................... 14

7.3 Hohmann Transfer (ΔV) Calculations (Kushal): ....................................................................... 19

8. Inclination Change Maneuver (Author: Kushal Shah & Jesus) ........................................................ 19

9. Total ΔV Calculations(Kushal Shah) ................................................................................................ 19

10. Mass Calculations( Kushal & Jesus).............................................................................................. 20

V. Power Calculation (Technical Details: Jesus Ramos) ....................................................... 21

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1. Initial Approach (Viral) ..................................................................................................................... 21

2. Choosing the Orbit (Jesus) ................................................................................................................ 22

VI. Rockets(Jesus & Omar) .......................................................................................................... 28

1. Choice of rockets (Omar and Jesus) .................................................................................................. 28

2. Rocket to be used in the launch (Jesus) ............................................................................................. 29

VII. Attitude Dynamics(Kushal & Carlos) ................................................................................... 30

1. Introduction (Carlos & Viral) ............................................................................................................ 30

2. Torque free attitude motion (Carlos) ................................................................................................. 30

3. Euler angles (Carlos & Mark) ........................................................................................................... 31

4. Analytical Model (Kushal, Mark & Omar) ...................................................................................... 31

5. Describing Rotation of S/C respect to Time domain (Kushal & Jesus) ............................................ 32

VIII. Attitude Control (Kushal & Mark) ................................................................................... 34

1. Introduction: (OMAR) ...................................................................................................................... 34

2. Strategy-Momentum Wheel ( Carlos & Omar) ................................................................................. 34

3. Analytical Model (Kushal & Jesus) ................................................................................................... 35

IX. Appendix (Team) ....................................................................................................................... 37

1. Parameters ......................................................................................................................................... 37

2. Equations (Mark, Omar, Viral).......................................................................................................... 37

3. Matlab Code (Jesus & Kushal) .......................................................................................................... 38

X. Work Cited(Team) ................................................................................................................. 51

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Table of Figures:

Figure 1: Shackleton Crater ........................................................................................................................ 6

Figure 2: J2000 - ECI Frame ...................................................................................................................... 9

Figure 3: Orientation of Moon and Earth ................................................................................................. 10

Figure 4: Kennedy Space Center .............................................................................................................. 10

Figure 5: Phasing Maneuver ..................................................................................................................... 12

Figure 6: Hohmann Transfer ..................................................................................................................... 13

Figure 7: Spacecraft 1 (Hohmann Transfer) ............................................................................................. 15

Figure 8: Spacecraft 2 (Hohmann Transfer) ............................................................................................. 16

Figure 9: Spacecraft 3 (Hohmann Transfer) ............................................................................................. 17

Figure 10: Spacecraft 4 (Hohmann Transfer) ........................................................................................... 18

Figure 11: Power Calculation Approach ................................................................................................... 21

Figure 12: Mirror Geometry ..................................................................................................................... 21

Figure 13: Orbit Choice around the Moon ................................................................................................ 23

Figure 14: Orbit around the Moon ............................................................................................................ 24

Figure 15: Atlas V Performance ............................................................................................................... 29

Figure 16: Body Reference Frame of the S/C ........................................................................................... 31

Figure 17: Theta/Euler's Angle at Function of Time ................................................................................ 33

Figure 18: Theta dot and Phi dot as function of Time .............................................................................. 33

Figure 19: Gyroscope ................................................................................................................................ 34

Figure 20: Momentum Wheel ................................................................................................................... 34

Figure 21: Angular Momentum Transfer and Angular Velocity Graphs ................................................. 36

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List of Tables:

Table 1: Bulk Parameters ............................................................................................................................ 8

Table 2: Pew chart for comparison of LEO vs. GEO ............................................................................... 11

Table 3: LEO Parameters .......................................................................................................................... 12

Table 4: Moon's Orbital Parameters ......................................................................................................... 13

Table 5: Hohmann Transfer Orbit Parameters .......................................................................................... 14

Table 6: Spacecraft 1 (Hohmann Transfer Data) ...................................................................................... 15

Table 7: Spacecraft 2 (Hohmann Transfer Data) ...................................................................................... 16

Table 8: Spacecraft 3 (Hohmann Transfer Data) ...................................................................................... 17

Table 9: Spacecraft 4 (Hohmann Transfer Data) ...................................................................................... 18

Table 10: Hohmann Transfer ΔV.............................................................................................................. 19

Table 11: Total Mission ΔV Data ............................................................................................................. 20

Table 12: Total Mission Mass Data .......................................................................................................... 20

Table 13: Parameters for Orbit around the Moon ..................................................................................... 24

Table 14: Illumination Power Coverage at the South Pole ....................................................................... 25

Table 15: Rocket Variations and Details .................................................................................................. 28

Table 16: Rocket Choice Comparison ...................................................................................................... 29

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II. Introduction(Viral) The idea of a colony on the moon has long been a favorite of science fiction writers and space

visionaries; however, with the data obtained from NASA’s Lunar Reconnaissance Orbiter (LRO) and

Chandrayaan-1, it seems possible. According to data from NASA's Lunar Reconnaissance Orbiter, ice

may make up as much as 22 percent of the surface material in Shackleton crater at the Moon's South

Pole (Figure 1). If humans are ever to inhabit the moon, the lunar poles may well be the location of

choice: Because of the small tilt of the lunar spin axis, the poles contain regions of near-permanent

sunlight, needed for power, and regions of near-permanent darkness containing ice — both of which

would be essential resources for any lunar colony [1].

The discovery of ice on the Moon has enormous implications for a permanent human return to

the Moon. Water ice is made up of hydrogen and oxygen, two elements vital to human life and space

operations. Lunar ice could be mined and disassociated into hydrogen and oxygen by electric power

provided by solar panels deployed in nearby illuminated areas or a nuclear generator. This hydrogen and

oxygen is a prime rocket fuel, giving us the ability to refuel rockets at a lunar "filling station" and

making transport to and from the Moon more economical by at least a factor of ten. Additionally, the

water from lunar polar ice and oxygen generated from the ice could support a permanent facility or

outpost on the Moon. The discovery of this material, rare on the Moon but so vital to human life and

operations in space, will make our expansion into the Solar System easier and reaffirms the immense

value of our own Moon as the stepping stone into the universe.

Figure 1: Shackleton Crater

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There are many concepts developed in order to utilize this source. Bill Stone, the founder of

Shackleton Energy Company, has proposed of mining these craters and in order to produce hydrogen

and oxygen, which could be used for as propellant. Other concept involves the development of

ecosystems to provide food, oxygen and recycle the available water. This project aims at exploring the

use of orbiting solar collectors to redirect Sun’s rays to these craters, eventually melting them.

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III. Modeling Assumptions (Carlos)

1. FOR THE MOTION OF THE MOON (CARLOS) o Lunar eclipses are ignored.

o The tilt of the moon is ignored.

o The motion of the Earth around the Sun is ignored in a first approximation.

o The moon is a perfect sphere, with uniform density, spinning around a fixed axis, parallel to the

ECI z-axis (point-mass)

o J2000 is taken to be ECI Frame for the mission (Inertial Reference Frame; Earth).

2. FOR THE MODELING (JESUS) o R2BP (Equations of Motion) serves as the starting point for a more complex study.

o Mass of the satellite is negligible to that of the Moon.

o No other forces act on the system except for gravitational forces (i.e. Ignore 3rd body

perturbations, atmospheric drag, and solar radiation pressure).

3. PARAMETERS FOR EARTH & MOON (MARK)

Bulk Parameters

Gravitational Parameter for Earth ( ) 398,600 [km3/s2]

Gravitational Parameter for Moon ( ) 4,902.77 [km3/s2]

Gravitational Constant (G) 6.67 x 10-11 [m3/kg*s2]

Mass of Earth ( 5.9736 x 1024 [kg]

Mass of Moon ( ) 0.07349 x 1024 [kg]

Radius of Earth (RE) 6371.01 [km]

Radius of Moon (RM) 1737.53 [km]

Escape Velocity of Earth 11.2 [km/s]

Escape Velocity of Moon 2.38 [km/s]

Earth-Moon Distance 384,405 [km]

Table 1: Bulk Parameters

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IV. Launch of Spacecraft from Earth to Moon (Technical

Details: Kushal Shah) 1. INTRODUCTION: (KUSHAL)

This section is used to describe our launch strategies and the frame it would be monitored during the

mission. AtlasV-431 will be initially launched from earth to low earth at the altitude of 1000 m orbit

using 26th March 2013 at 15:04 (CST). Then, all the s/c will perform phase change maneuvers in the LEO

orbit to position in desired location in such way that they end up at the desired positions at the time of

launch for Hohmann transfer to the Moon. These positions were solved using equations, which describes

s/c motion in LEO at all time and the Moon’s motion in its orbit at all time, described in Appendix. After

positioned correctly in LEO, they will perform Hohmann transfer according to their launch position and

time window as chosen to meet the designed locations in the orbit around the moon. These designed

locations were planned and calculated through the MATLAB code written by team member (Jesus

Ramos). Then, they will perform small inclination change maneuvers to achieve positions designed to

meet the mission requirements.

2. EARTH CENTRIC INERTIAL (ECI) FRAME AND COORDINATE SYSTEM DESCRIPTION (AUTHOR: KUSHAL & VIRAL) This section is used to explain how is our ECI frame is defined

to track the satellite during the mission. XY-plane is the plane of the

Earth’s Orbit at the reference epoch. X axis is defined in such a way

that the ascending node of the moon’s plane is approximately

226.75 Degrees. In other words, x-axis is out along ascending node

of plane of the Earth's orbit and the Earth's mean equator at the

reference epoch of J2000.0. Y-axis is perpendicular to 90 degrees

east to defined x-axis. Z- Axis is perpendicular to the XY-plane

defined being positive in sense of Earth’s North Pole at the

reference epoch of J2000.0 Figure 2: J2000 - ECI Frame

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3. SELECTING LAUNCH SITE (AUTHOR: KUSHAL ) Consumption of fuel and the simplicity of transfer are the main factors what were taken into the

consideration when selecting a launch site. Then, looking at the figure 4, it was decided that the rocket

should be launched from the latitude of 28 degrees north or south. Kennedy Space Center is located at

80.6o West longitude and 28.6o North latitude, which is approximately same as moon’s orbital

inclination. Furthermore, Apollo 11, first spaceflight that landed first humans on the moon, was

launched from the Kennedy Space Center. By choosing Kennedy Space Center as the mission launch

site, the value of delta V needed was minimized for transfer since it will eliminate to maneuver to

change the plane to match the moon’s orbital plane.

4. SENDING THE SPACECRAFT TO ITS FIRST ORBIT AROUND THE EARTH (AUTHOR: KUSHAL)

4.1 Choosing the Orbit (LEO vs. GEO):

A. Low Earth Orbit (LEO) – Low earth orbit is defined as an orbit below an altitude of

approximately 2,000 kilometers above sea level. One of the major advantages of low earth

orbit is that it allows us to communicate with spacecraft after its initial launch since it is

very close to the earth.

B. Geostationary Earth Orbit (GEO) – GEO is a circular orbit 35,786 kilometers above the

Earth's equator. This orbit is unique because it follows the direction of the Earth's rotation at

all the times. However, this orbit requires special permit and is exposed to Van Allen belt

opposed to Leo which can cause damage to our spacecraft. The radiation and the

Figure 3: Orientation of Moon and Earth Source: http://visibleearth.nasa.gov/

Figure 4: Kennedy Space Center

Source: www.ksc.nasa.gov

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spacecraft’s distance from the Earth can significantly reduce the ability of communication

with the spacecraft. This can be problematic.

C. Pew Chart to Compare LEO and GEO -

Parameters Weight (1-5)

LEO GEO (1-5) Score (1-5) Score

Communication w/ SC 5 5 5*5=25 Pro 1 5*1=5 Con

Cost of Initial Launch from Earth

5 5 5*5=25 Pro 2 5*2=10 Con

Distance from Moon 2 2 2*2=4 Con 5 2*5=10 Pro

Mass of Fuel for ΔV1 3 2 3*2=6 Con 4 3*4=12 Pro

Total Score 60 37

Table 2: Pew chart for comparison of LEO vs. GEO

D. Decision - For the mission communication with the s/c and cost of initial lunch is more

important. Based on pros and cons from the pew chart based and the mission priorities for it

was decided that the rocket, Atlas V will launch the s/c from earth to LEO. Then spacecrafts

will perform Hohmann orbital transfer to enter into the orbit around moon which will be

used to provide a constant illumination power to the south pole of the moon.

4.2 Choosing the Altitude from Sea Level to LEO Orbit: Next step for the mission was to decide what altitude can be used and will be cost

effective. The conclusion was made that altitude for low earth orbit must be greater than 300 km

to avoid the atmospheric drag and less than 2000 km to avoid entering in the region of Van Allen

Radiation Belt which may damage the spacecraft. Based on these lower and upper limits, it was

decided that the altitude for the LEO orbit will be 1000 km, which is still close to earth for the

communications with the spacecraft.

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4.3 Calculations for LEO Orbit, after Arrival in the Orbit from the Launch (Mark & Kushal):

Parameters / Orbital

Elements Equations / Assumptions Value

Radius of the Orbit / Semi

Major Axis (a) Altitude + Radius of Earth 7.3781e+03 km

Velocity

7.3502 km/s

Eccentricity Circular orbit 0

Ascending Node (Ω)

(Degrees) Launched such that it aligns with Moon’s Ω 226.75o

Inclination (i) True – 5.080 o

Assumed - 5.26 o to align with moon’s plane 5.26o

True Anomaly (Ө) Launched such way that it ends up on top of the orbit 90o

Table 3: LEO Parameters

5. PHASING MANEUVERS (AUTHOR: OMAR & MARK) All five spacecraft will be launched together from the

ground to the altitude of 1000km (LEO). From there they will be

separated within the same orbit using phase maneuvers. A phase

maneuver is a two-impulse Hohmannn transfer from and back to

the same orbit. The mission requires us to place five spacecraft

strategically in an orbit around the moon. Since all the spacecraft

are going to be in the same exact orbit around the moon, they

need to be phase maneuvered so that they are evenly spread

apart in the orbit to achieve position of desired locations on the

orbit around the moon. Figure 5: Phasing Maneuver

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6. DESCRIBING MOON’S ORBIT (MARK & OMAR) This table shows the parameters and its values that describe moon’s orbital motion around the

moon. These values were used to design the locations of s/c in the orbit around the moon and in the low

earth orbit to decide how to orient them initially for the correct position at the time of Hohmann transfer

launch. As mentioned earlier. These equations and the MATLAB code are listed in Appendix. The

positions and the orbital elements of moon in the J2000 were taken from the JPL solar system dynamics

ephemerides section.

7. THE HOHMANN TRANSFER ( KUSHAL )

7.1 The Hohmann Transfer Concept (Author: Viral): The diagram shows a Hohmann transfer orbit to bring a

spacecraft from a lower circular orbit into a higher one. It is one half of an elliptic orbit that touches both the lower circular orbit and the higher circular orbit (moon’s orbit around the earth).

The spacecraft will perform ∆V1 maneuver at the apogee of an elliptical orbit. The ∆V1 can be found by the following equation:

Where, r1 = Radius of LEO orbit from ECI Center

Moon’s Orbital Elements on S/c Arrival Date - 2013-Apr-15 22:22:00

Semi – Major Axis 381,454.713 (km)

Eccentricity (e) 0 (by assumption)

Inclination (i) 5.26o

Ascending Node (Ω) (Degrees) 226.75o

Mean Motion (n) 1.545E-04 (Degrees/sec)

True Anomaly (Ө) 180.00 o

Table 4: Moon's Orbital Parameters

Figure 6: Hohmann Transfer

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r2 = the distance between earth and moon When the spacecraft has reached its destination orbit (at perigee) the gravitational field of the moon

captures it, and the spacecraft starts rotating around the moon. Another maneuver ∆V2 is performed in order to achieve the desired orbit around the moon. We will use the similar approach for all five spacecraft.

Hohmann’s Transfer Orbit From LEO (7378km) to MOON Elements

Parameters Equations Values

Semi – Major Axis

1.9589e+05 (km)

Eccentricity (e) 0.9623

Inclination (i) Similar to LEO inclination 5.26 o

Ascending Node (Ω) Similar to LEO Ascending Node 226.75 o

Mean Motion (n)

7.2820e-06

True Anomaly (Ө)

At perigee Departure Angle (Fastest Speed) 0 o

True Anomaly (Ө)

At apogee Arrival Angle (Lowest Speed) 119.8378j

Hohmann Transfer Time (TH)

119.8378 (Min) = 5hrs

Table 5: Hohmann Transfer Orbit Parameters

7.2 Hohmann Transfer Positions and Timing: (Mark, Kushal & Omar) Following is the position and time data for all four spacecraft. Maneuver beginning refers to the

time of launch from LEO and maneuver end refers to the time when spacecraft will enter the moon orbit.

The positions are with respect to ECI frame. These were obtained through MATLAB code written by

Kushal Shah. The positions and orbital elements of moon in the J2000 at given/desired time were taken

from the JPL solar system dynamics ephemerides section.

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7.2.1 Spacecraft 1 (Hohmann Transfer Data):

Maneuver Beginning

Date: 03/26/13

Time: 19:00

Maneuver end

Date: 3/31/13

Time: 19:00

Moon’s Orbit’s Coordinates in ECI

ω=N/A ω=N/A

Ө = 294.52o Ө = 0o

Xmi (km) -3.63E+05 Xmf (km) -2.63E+05

Ymi (km) 1.22E+05 Ymf (km) -2.80E+05

Zmi (km) -3.21E+04 Zmi (km) 0.00E+00

S/C1 Coordinates of Hohmann transfer Orbit in ECI

ω=180 o ω=180 o

Ө = 0o Ө = 180o

Xs/ci (km) 5.06E+03 Xs/cf -2.63E+05

Ys/ci 5.37E+03 Ys/cf -2.80E+05

Zs/ci 0.00E+00 Zs/cf -4.32E-12

S/C Position in LEO 180o N/A

S/C Position in LEO at the Initial Launch 90o -

Time From Initial Launch: 3.94hr 1

Table 6: Spacecraft 1 (Hohmann Transfer Data)

Figure 7: Spacecraft 1 (Hohmann Transfer)

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7.2.2 Spacecraft 2 (Hohmann Transfer Data):

Maneuver Beginning Date: 03/31/13 Time: 15:36

Maneuver end Date: 04/05/13 Time: 15:36

Moon’s Orbit’s Coordinates in ECI

ω=N/A ω=N/A

Ө = 358.78o Ө = 64.26o

Xmi (km) -2.74E+05 Xmf (km) 1.37E+05

Ymi (km) -2.70E+05 Ymf (km) -3.58E+05

Zmi (km) -1.37E+03 Zmi (km) 3.17E+04

S/C1 Coordinates of Hohmann transfer Orbit in

ECI

ω=243.52 o ω=243.52o

Ө = 0o Ө = 180o

Xs/ci (km) -2.54E+03 Xs/cf 1.32E+05

Ys/ci 6.90E+03 Ys/cf -3.60E+05

Zs/ci -6.05E+02 Zs/cf 3.15E+04

S/C Position in LEO 243.52o N/A

S/C Position in LEO at the Initial Launch 107o -

Time From Initial Launch: 121.5419 2

Table 7: Spacecraft 2 (Hohmann Transfer Data)

Figure 8: Spacecraft 2 (Hohmann Transfer)

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7.2.3 Spacecraft 3 (Hohmann Transfer Data):

Maneuver Beginning

Date: 04/01/13

Time: 00:49

Maneuver end

Date: 04/06/13

Time: 00:49

Moon’s Orbit’s Coordinates in ECI

ω=N/A ω=N/A

Ө = 8.39o Ө = 73.87o o

Xmi (km) -2.20E+05 Xmf (km) 1.95E+05

Ymi (km) -3.15E+05 Ymf (km) -3.30E+05

Zmi (km) 5.14E+03 Zmi (km) 3.39E+04

S/C1 Coordinates of Hohmann transfer Orbit in

ECI

ω=253.25o ω=253.25o

Ө = 0o Ө = 180o

Xs/ci (km) -3.67E+03 Xs/cf 1.91E+05

Ys/ci 6.37E+03 Ys/cf -3.32E+05

Zs/ci -6.48E+02 Zs/cf 3.37E+04

S/C Position in LEO 253.25o N/A

S/C Position in LEO at the Initial Launch 103 -

Time From Initial Launch: 139.13 hrs 3 Table 8: Spacecraft 3 (Hohmann Transfer Data)

Figure 9: Spacecraft 3 (Hohmann Transfer)

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7.2.4 Spacecraft 4 (Hohmann Transfer Data):

Maneuver Beginning Date: 04/02/13

Time: 3:47

Maneuver end Date: 4/07/13 Time: 3:47

Moon’s Orbit’s Coordinates in ECI

ω=N/A ω=N/A

Ө = 18.01o Ө = 83.51o

Xmi (km) -1.64E+05 Xmf (km) 2.47E+05

Ymi (km) -3.47E+05 Ymf (km) -2.92E+05

Zmi (km) 1.09E+04 Zmi (km) 3.50E+04

S/C1 Coordinates of Hohmann transfer Orbit

in ECI

ω=261.12o ω=261.12o

Ө = 0o Ө = 180o

Xs/ci (km) -4.51E+03 Xs/cf 2.35E+05

Ys/ci 5.80E+03 Ys/cf -3.02E+05

Zs/ci -6.68E+02 Zs/cf 3.48E+04

S/C Position in LEO 261.12 N/A

S/C Position in LEO at the Initial Launch 95.16 -

Time From Initial Launch: 156.73 hrs 4

Table 9: Spacecraft 4 (Hohmann Transfer Data)

Figure 10: Spacecraft 4 (Hohmann Transfer)

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7.3 Hohmann Transfer (ΔV) Calculations (Kushal): The calculation of ΔV for Hohmann transfer is described in the table below with the equations

used during the computation.

(Where r1 = Radius of LEO orbit from ECI Center & r2= distance between earth and moon):

Parameters Equations Value

Velocity at given True Anomaly of Transfer Orbit

-

Required ΔV1 To transfer from LEO to Hohmann Transfer Orbit

2.9462 km/s

Required ΔV2 To transfer from Hohmann Transfer Orbit to Moon’s Orbit

0.8207 km/s

Total Hohmann Transfer ΔVt 3.7669 km/s

Table 10: Hohmann Transfer ΔV

8. INCLINATION CHANGE MANEUVER (AUTHOR: KUSHAL SHAH & JESUS) This maneuver is known as orbital inclination change maneuver, also known as plane change

maneuver. This plays huge role in our calculations because it reduced the number of s/c used from five to

four because small inclination change allowed us to focus more of Sun’s energy than initially at 90o

inclination. It is done at the orbital node of the orbit around the moon to change its inclination little more

than 90 degrees. While performing this maneuver we are keeping the speed same just changing its

direction to achieve designed inclination. Using the equation below, we achieved that our ΔV for plane

change is approximately 0.0933 km/s.

Equation Follows: ; where f = true anomaly.

9. TOTAL ΔV CALCULATIONS(KUSHAL SHAH) Next, total ΔV used for mission was computed by summing all the individual ΔV for spacecraft

form different maneuvers and transfer orbit. All these calculations were done using the common

equations described previously in each section of maneuvers and integrated in the MATLAB code

written by Kushal Shah. Individual ΔV data for each s/c are described in the table below. The total ΔV

for the mission is 15.945 km/s.

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S/C 1 S/C 2 S/C 3 S/C 4

Phase Change (Degrees) 0 17 13 5.6

Phase Change ΔV (km/s) 0 0.2429 0.1836 0.0774

Hohmann Transfer ΔV (km/s) 3.7669 3.7669 3.7669 3.7669

Plane Change ΔV (km/s) 0.0933 0.0933 0.0933 0.0933

Total ΔV (km/s) 3.8602 4.1031 4.0438 3.9376

Total Mission ΔV (km/s) 15.9447

Table 11: Total Mission ΔV Data

10. MASS CALCULATIONS( KUSHAL & JESUS) Once the data table of total ΔV for the mission was obtained, then total mass for the mission was

calculated and individual s/c mass along with the structure mass and payload mass using the equation below.

It was assumed that each spacecraft will carry 100kg of payload and has 500kg of structure mass. All the individual mass and the total mass are listed in the table below. The total mass for the four s/c combined is 12,201 kg, which will be payload mass for the Atlas V-431.

S/C 1 S/C 2 S/C 3 S/C 4

Total Fuel Consumed (kg) 2296 2597 2521 2388

Structure Mass (kg) 500 500 500 500

Payload Mass (kg) 100 100 100 100

Total Mass (kg) 2896 3197 3121 2988

Total Fuel for the Mission (kg) 9,801

Total Mass for the Mission (kg) 12,201

Table 12: Total Mission Mass Data

Discussion: The mission constraint was maximum usage of 500kg of fuel for each spacecraft. However,

to meet that requirement, Atlas V-431 needs to be launched to Geostationary Earth Orbit (GEO) which

requires special permit and increases the rocket launch price greatly. In short, this design of making

LEO parking orbit is more efficient than making GEO parking orbit for the mission because it will allow

smooth communication with s/c before Hohmann transfer while in LEO, because LEO is closer and not

exposed to Van Allen belt. The alternative can be bi-elliptical Hohmann transfer which can be cheaper

since the ratio of distance of moon from earth and LEO altitude is greater than 11.94.

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V. Power Calculation (Technical Details: Jesus Ramos) 1. INITIAL APPROACH (VIRAL)

Following is an example of an orbit around the moon:

Figure 11: Power Calculation Approach

The orbital plane of a spacecraft is perpendicular to the sun rays. The sun rays are considered to

be parallel since they are coming from relatively infinite distance. Since the given power is 1 kW/m2 and

the given area for the mirror is 100 m2, each mirror will provide 100kW if the sun rays are directly

perpendicular to the mirror. In our case, the angle of incidence ψ is constantly changing as the spacecraft

orbits around the moon.

Figure 12: Mirror Geometry

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- Computing the position of spacecraft respect to South Pole

Using the rotation matrices, we computed the position vector of spacecraft in the frame

attached to the moon center. Next, we computed the position of South Pole respect to the frame

attached to the center of moon. Using vector subtraction we computed the position vector of s/c

respect to South Pole.

The Computation:

1. Computing Position of S/C in the frame attached to center of the earth.

2. Computing Position of South Pole in the frame attached to center of the earth.

3. Computing Position of S/C respect to the South Pole.

2. CHOOSING THE ORBIT (JESUS)

Before the orbits for the space crafts were chosen, a few assumptions were made. The ECI frame was

fixed to the moon’s center with the moon as point mass. The problem was simplified to a two body

problem with the moon and one of the spacecraft. This was a valid assumption since the other space

crafts’ mass will not affect their orbits due to their extremely low mass. The moon was titled to the 5.14

degrees; the South Pole was fixed at the point of where the axis is at the bottom.

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Given that, the following equation based on newton’s law was used

(Orbit equation)

where the orbital elements are the semi-major axis (a), eccentricity (e), inclination (i), argument

of periapsis (ω), true anomaly (θ), longitude of ascending node (Ω).

Of the four types of possible orbitals – circle, ellipse, parabola, and hyperbola- an elliptical orbit was

chosen because with the parabola and hyperbola they were unbounded orbits, allowing the spacecraft to

escape the moon’s gravity. Of the bounded orbitals, only the elliptic orbital allowed for a longer time

near the aposis of the orbit. This allowed for the spacecraft to stay longer in the illuminated region.

A matlab script was written that plotted the orbit given the radius of apogee and perigree, calculated the

time that the spacecraft spent in the region where the sunlight can hit the mirror and reflect to the South

Pole. The script also plotted the power that it received from that region as well.

Steps in choosing the orbit

1. Set the radius of perigee to a distance away from the moon’s surface

a. The radius was set to 1790 km which was 52 km away from the surface, reasonable distance

2. Set the radius of the apogee

3. Adjust the values of the inclination, argument of periapsis, and longitude of ascending node

4. Check for the power output and time spent in the region.

5. Repeat step 3, and then step 2 followed by 3.

Figure 13: Orbit Choice around the Moon

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6. Once an orbit was found that satisfied the objectives, the orbital elements were slightly

changed to improve the time and power

7. Check if whether if 4 or 5 spacecraft could be placed on that one orbit and still achieve the

objectives. Otherwise, another orbit would need to have been found.

The orbit that allowed the spacecraft to spend most of the time in the region was with the

following orbital elements

A table was created given the results of this orbit and the conclusion was that at least 4 spacecraft

can be used to meet the objectives. They would be placed into orbit starting that the radius of periapsis at

specific times to position them correctly.

Timeline At t = 0 Insert first SC at Rp At t =1.20187 hr 1st SC is at 125.3 degrees At t = 2.789869, 2nd SC is inserted at Rp At t= 3.991739, 2nd SC is at 125.3 degrees At t= 5.19361, 3rd SC is inserted at Rp At t = 6.39598, 3rd SC is at 125.3 degrees At t=7.597351, 4th SC is inserted at Rp At t = 8.799221, 4th SC is at 125.3 degrees

Orbital Elements for The Orbit

Semi-major axis (a) [km] 5439.94

Eccentricity (e) 0.67

Inclination (i) -90

Argument of Periapsis (ω) -90

True Anomaly (θ) 0o <= θ <=360o

Longitude of Ascending Node (Ω) 0

Table 13: Parameters for Orbit around the

Figure 14: Orbit around the Moon

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In the graph below, the spacecraft operates between 125.3 to 234.7 degrees (the middle red

line is the 180 degree mark) and at t=0 at the periapsis between 1.20-8.80 hours.

The power that was achieved by one spacecraft during that angle range ranged from a

minimum of 37.95% to a maximum of 39.94% power of the total necessary. To avoid complicating the

computation of theta, the power was set to the minimum and the angle was kept constant. Below is the

power that was kept constant at the minimum power and the theta values for the spacecraft.

TIME(HR) 0-0.386 0.386-1.201 1.201-2.789 2.789-5.193 5.193-7.597 7.597-10.001

S/C #1 (% of Total Output) 37.95 37.95 37.95 37.95 37.95 0

S/C #4 (% of Total Output) 37.95 37.95 37.95 37.95 0 37.95

S/C #3 (% of Total Output) 37.95 37.95 37.95 0 37.95 37.95

S/C #2 (% of Total Output) 37.95 0 0 37.95 37.95 37.95

Total Power Output (% of Total Needed) (Total needed

is %100) 151.8 113.85 113.85 113.85 113.85 113.85

if 1 s/c failed power supplied (% of total needed) (Total

needed is %75) 113.85 75.9 75.9 75.9 75.9 75.9

Table 14: Illumination Power Coverage at the South Pole

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Below is a plot of the four spacecraft at t = 10 hr where the true anomalies for the four spacecraft were:

0 , 151.68 , 178.15 , and 203.43

Discussion

Although there are countless of different other possible orbits to have chosen from, this orbit and

having four spacecraft on it works. It is able to provide more than 200 Kw at all times and still provide

more than 150 Kw if one of the spacecraft fails as shown in the table.

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VI. Rockets 1. Choice of rockets (Omar and Jesus)

From the calculations that were done, the total mass of the four spacecraft (including the fuel,

structural, and payload mass) was about 12,201 kg. In the table below are three highly successful rockets

in sending spacecraft into space and their specifications that looked promising.

Rocket Atlas V Delta II Delta IV

Function EELV/ Medium-Heavy

Launch Vehicle

Medium-Heavy

Launch Vehicle

Medium-Heavy

Launch Vehicle

Provider United Launch Alliance United Launch

Alliance

United Launch

Alliance

Mass 334,500 kg (737,400 lb)

151,700 – 231,870

kg (334300 –

511180 lb)

249,500 – 733,400 kg

(550,000-1,616,800 lb)

Stages 2 2 or 3 2

Payload to

LEO

9,370-29,400 kg (20,650 –

64,820 lb)

2,700 – 6,100 kg

(5,960 – 13,400 lb)

8,600 – 22,560 kg

(18,900 – 49,740 lb)

Cost $90 - $110 Millions $45 – $60 Millions $140 – $170 Millions

Table 15: Rocket Variations and Details

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And some of their pros and cons are listed below in the following table.

Pros Cons

Atlas V

• Carries the heaviest payload

• Cheaper that the Delta IV

• Only 1 partially failed launch (97.14%

success rate)

• Not the cheapest rocket to use

• Does not have the highest

success rate

Delta II

• Cheapest rocket

• Commonly sent into space (149 successful

launches

• Only 2 partially failed launches (98.67%

success rate)

• Most successful

• Carries about half the amount

of the other rockets

• May not be able to carry the

amount of satellites that we

want it to

Delta IV

• Carries a high payload

• Only 1 partially failed launch (95.24%

success rate)

• Most expensive

• Least successful

Table 16: Rocket Choice Comparison

2. ROCKET TO BE USED IN THE LAUNCH (JESUS) The rocket that was chosen to launch the four spacecraft was the Atlas V-431 because of its high

success rate, and its overall cost (~$110 million vs 2 x $60 million using Delta II vs $140 million using

Delta IV), and its capability to send a payload of 15130 kg to a low Earth orbit (at 400 km). Since the

LEO for this mission was 1000 km, this rocket with a higher payload was chosen to make sure the

spacecraft could be inserted into that 1000 km. Below is the performance chart for the Atlas V rockets.

Figure 15: Atlas V Performance

Source: http://www.ulalaunch.com/site/pages/Products_AtlasV.shtml

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VII. Attitude Dynamics (Kushal & Carlos) 1. INTRODUCTION (CARLOS & VIRAL)

The orientation of spacecraft in space is called attitude and the kinematic equations related to

geometry in free space falls into the category of attitude dynamics. A rigid body can be viewed as a

system of particles where the relative distances between particles are fixed. The position of a rigid body

is defined by any three points on it, not in the same straight line. The motion of a rigid body can be

described by a translation of some reference point O, plus a rotation about some axis through O.

The goal of attitude determination is to determine the orientation of the spacecraft relative to either

an inertial reference frame or some specific object of interest, such as the moon at all time during the

mission. To do this, we must have available one or more reference vectors, i.e., unit vectors in known

directions relative to the spacecraft. The purpose for our project is to point our instrument/mirror at the

South pole of the moon. The orientation of the spacecraft needs to be stabilized to some desire attitude.

To achieve this, we must ask ourselves:

-where are we?

-where do we want to be?

-how do we get there?

2. TORQUE FREE ATTITUDE MOTION (CARLOS) The disturbance torques are typically very small, so torque free motion is a good approximation

over short periods of time, and can provide us with an understanding of the dominant behavior of the

natural attitude motion of a spacecraft.

There are special cases of axisymmetric spacecraft, which allow us to obtain a closed-form

solution of the attitude motion. An axisymmetric body is one in which two of the principal inertias are

equal, for example, a cylinder of uniform density.

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3. EULER ANGLES (CARLOS & MARK) The most commonly used sets of attitude parameters are the Euler angles (Φ,θ,Ψ). For our design,

we set ω1 and ω2 to zero and solving for ω3. In other words, our spacecraft only moves about the third

axis. These equations were achieved based on general rotation of any space craft using the general (3-1-

3) attitude matrix and the cue symmetric Ω (ω x) matrix.

4. ANALYTICAL MODEL (KUSHAL, MARK & OMAR) All these concepts, described above were used to conclude the

determination of the frame attached to the body of the spacecraft for

the attitude determination of the spacecraft at all time for this mission.

Geometry of the spacecraft is designed such way that it is

axisymmetric about all three axes. The frame attached to the body of

the spacecraft is (O,b1,b2,b3). It is able to rotate about its major axis

that is b3 direction since for our model Izz>Iyy>Ixx. (figure). As

mentioned, that it is symmetric about its all axis, thus it is principle

body frame and has the inertia tensor in the form of:

The rotation of spacecraft from its inertial orientation is denoted as phi (ϕ) and the rate it needs to

rotate to achieve desired attitude is the rate of change of ϕ respect to time. There is no other rotation due

to only need of one axis spin for this mission. This lead to required determination of attitude matrix

between body fixed frame at given time and inertial orientation of the body, which is shown below.

Figure 16: Body Reference Frame

of the S/C

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Using attitude matrix and the assumptions of the rotation (ie. it rotates only about phi (ϕ)), we

attained the simple relation between the orientation of the spacecraft and the rotation about third axis is

same. That is, . In this simplified equation, ωz is the rotation needed at given time in the orbit

around the moon.

Furthermore, using the Euler’s equations in the principle body frame described below and the

assumption that it only rotates about third axis, we concluded that the rotation about third axis will be

constant, if initial angular velocity applied, for all time unless the momentum is transferred or changed

through control system.

Euler’s Equation: ; where I and ω is expressed in body fixed frame.

Euler’s simplified equation in Principle Body Frame:

Next, using the designed desired angular veolcity to achieve 200 kW at the south pole of the moon, it

was decided that desired spin about third axis ωz is the same as the rate of change of true anomaly

respect to time, that is . This lead to relationship that the true anomaly (ϴ) = Euler’s angle (ϕ) for

the orbit around the moon.

5. DESCRIBING ROTATION OF S/C RESPECT TO TIME DOMAIN FOR THE ORBIT AROUND THE MOON(KUSHAL & JESUS) As mentioned earlier that the true anomaly (ϴ) = Euler’s angle (ϕ) for the orbit around the moon and

the rate of change of true anomaly respect to time, that is . This section shows the diagram of the

behavior of Euler’s angle and its time derivative variation respect to time. This section includes the

graphs of Euler’s angle and its time derivative variation vs. time.

5.1 Theta as Function of Time: This relationship was attained using simple orbital equations and Kepler’s equation to

describe motion in time. These equations are in the appendix section of this report.

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5.2 Theta Dot as Function of Time:

This relationship was attained using derivative of simple orbital equations and Kepler’s

equations. These equations are in the appendix section of this report. The approach was to use chain

rule, which follows

where ϵ is the eccentric anomaly from Kepler’s equations.

Figure 17: Theta/Euler's Angle at Function of Time

Figure 18: Theta dot and Phi dot as function of Time

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VIII. Attitude Control (Kushal & Carlos) 1. INTRODUCTION: (OMAR)

The rotation of our spacecraft is crucial to our mission, so we need a

method of controlling it. In order to do this we will utilize control moment

gyroscopes (CMGs). Control moment gyroscopes function by using

spinning rotors and motorized gimbals. The gimbals tilt the rotor’s angular

momentum, which produce gyroscopic torque, and rotates the spacecraft.

For our basic satellites we will use single-axis CMGs; we will need at

least 3 of them to fully control the attitude of our spacecraft.

2. STRATEGY-MOMENTUM WHEEL ( CARLOS & OMAR) Most of the spacecraft contain one or more spinning rotors to provide stability of a desired

orientation or attitude of the vehicle. For our mission, we will use momentum wheels, one of the efficient strategies, which contain electric motor attached to spacecraft. This momentum wheel is free to rotate about one axis. For this mission, it is free to rotate about b3 - 3rd axis direction. This wheel will be controlled and managed by the spacecraft’s onboard attitude control computer. Momentum wheel is used when the external body of the satellite must spin slowly to accomplish its mission while achieving needed the stabilization. The angular momentum of the wheel will be driven by a motor attached to the satellite and the satellite rotates in opposite direction based on the Newton’s third law of action - reaction. In this manner, it is also known as momentum transfer technique, which not increases nets momentum of the system. In the end, there will be no net momentum increase occurs when an adjustment in rotor speed is made. This is the strategy of the mission to control desired attitude control of the spacecraft to focus Sun’s energy to Moon’s South Pole.

For the simplification purposes, principal axes of the electric motor are aligned with the satellite

principal axes. Thus, the momentum wheel is axisymmetric, and the center of mass of the momentum

Figure 19: Gyroscope

Source: http://www.nasa.gov

Figure 20: Momentum Wheel

Source: http://www.nasa.gov

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wheel is placed at the center of mass of the satellite. As result, the momentum wheel is constrained to

move with the rotational motion of the satellite and principal axes of the body attached frame remain

principal, which provides great simplification to the problem.

3. ANALYTICAL MODEL (KUSHAL & JESUS) For this mission, the satellite is modeled as a platform, denoted by P, for this section and the

momentum wheel is modeled as action base, denoted by R. Furthermore, the angular momentum is

described by multiplication of Inertia tensor and the angular velocity of the body. That is, H = [I]ω.

As has been noted, both platform and the reaction base are aligned with the principle axis. Thus, the

form of the inertia tensor for the platform and the action base is described below.

The angular velocity of the platform will be measured using sensors such as MEM devices and

gyroscopes, and the angular velocity of the action base will be controlled through feedback controller.

The feedback control will provide needed momentum transfer to platform based on desired positions

through sensors attached to platform. However, during the momentum transfer, there is no net

momentum increase or decrease so it can be modeled as total momentum is equal to zero.

Using this model, the required angular velocity that momentum wheel needed to output to satellite

were computed. This computation was done in order to observe the control of the attitude of the

spacecraft to achieve desired attitude orientation. Since, the body frame only rotates about third axis

(wx, wy = 0) and both the body and the momentum wheel are axisymmetric about principle body frame,

the equation simplifies to:

Where, ω3 [P] and ω3 [R] is not equal and are in different directions about same axis.

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These are the following graphs based on the Izz P = 5 (kg* m^2) and Izz R = 3 (kg* m^2) of the

mission.

Figure 21: Angular Momentum Transfer and Angular Velocity Graphs

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IX. Appendix (Team)

1. PARAMETERS Semi Major Axis a

Eccentricity e

True Anomaly f, θ

Mean Motion n

Mean Anomaly Me

Eccentric Anomaly E, ϵ

Angular veolcity ωi

2. EQUATIONS (MARK, OMAR, VIRAL) 2.1 Relevant Phasing Maneuver Equations

Relevant Phasing Maneuver Equations

Period of Orbit 1

Semimajor Axis of Orbits

Angular Momentum

Eccentricity Anomaly

Speed found by Angular

Momentum Formula

Velocity Changed Required

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2.2 Choosing the Initial and Final Positions:

2.3 Kepler’s Equations:

3. MATLAB CODE (JESUS & KUSHAL) This appendix contains the MATLAB codes that were used to compute the orbit’s parameters,

power, and other values that were used in the report. A description of what the code does is found in the

code.

3.1 Orbital Analysis (Authors: Jesus Ramos) ...................................................................................... 39

3.3 Kepler’s Equation (Author: Jesus Ramos) ..................................................................................... 44

3.4 To Calculate Theta and Theta dot as Function of Time. (Author: Kushal Shah) .......................... 45

3.5 Calculating Positions of Hohmann Transfer (Author: Kushal Shah) ............................................ 46

3.6 Calculating ΔV for Hohmann, Phase Transfer and Plane Change (Author: Kushal Shah) ........... 48

3.7 Calculating initial and final positions (Author: Kushal) ................................................................ 50

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3.1 Orbital Analysis (Authors: Jesus Ramos) %MAE 146: Astronautics %Properties of the moon m_radius = 1738; %km mu = 4902.7779; %km3*s^-2 %Properties of the Earth e_radius = 6353; %km emu = 393600; %km3*s^-2 %Orbital Elements % Input is in degrees Ra = 5.23*m_radius; %Radius Apogee [km] Rp = 1.03*m_radius; %Radius Perigee [km] e1 = (Ra-Rp)/(Ra+Rp); %eccentricity true_a = 180; %true anomaly inc = -90; % inclination angle omega = 0.0001; %Longitude of ascending node angle w = -90; %argument of periapsis angle n=500; %number of elements %--------------------------------------------------------------- a = (Rp + Ra)/2; %Calculates the semimajor axis "a" [km] %%Plots the moon in XYZ coordinates subplot(2,2,1) [x y z] = sphere(); sphere = surf(m_radius*x, m_radius*y, m_radius*z); set(sphere, 'FaceAlpha',0.15); colormap gray; daspect([1 1 1]); xlabel('X axis'); ylabel('Y axis'); zlabel('Z axis'); hold on %Angle from 0 to 2Pi, and from 0 to 360 theta = (0: 2*pi()/n: 2*pi()); thetadeg = (0:360/n:360); %Plots the point of the SOUTH POLE x_south = 0; y_south = 0; z_south = -m_radius; plot3(x_south, y_south, z_south, 'ro'); %%Plots the orbit r1 = a*(1-e1^2)./(1 + e1*cos(theta)); %True anomaly r_ta = a*(1-e1^2)./(1 + e1*cosd(true_a)); [Xa, Ya, Za] = pol2cart(true_a/180*pi(), r_ta, 0); z0 = zeros(size(theta)); %%Converts from polar to cartesian coordinates [X1,Y1, Z1] = pol2cart(theta, r1, z0);

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%Rotation Matrices R_omega = [cosd(-omega) sind(-omega) 0; -sind(-omega) cosd(-omega) 0; 0 0 1]; %about the 3rd axis R_inc = [1 0 0; 0 cosd(-inc) sind(-inc); 0 -sind(-inc) cosd(-inc)]; %about the 1st axis R_w = [cosd(-w) sind(-w) 0; -sind(-w) cosd(-w) 0; 0 0 1]; %about the 3rd axis %Result if after the rotation of all three angles result = R_omega*R_inc*R_w*[X1; Y1; Z1]; %Result after the rotation for the true anomaly result_anomaly = R_omega*R_inc*R_w*[Xa; Ya; Za]; %%Draws the EQUATOR eq = 0; R_eq = [1 0 0; 0 cosd(-eq) sind(-eq); 0 -sind(-eq) cosd(-eq)]; %about the 1st axis [X0, Y0] = pol2cart(theta, m_radius); result_eq = R_eq*[X0; Y0; z0]; subplot(2,2,1) plot3(result_eq(1,:),result_eq(2,:),result_eq(3,:), 'r-.'); %Gets each value (x, y, z) and plots the orbit xp = result(1,:); yp = result(2,:); zp = result(3,:); plot3(xp, yp, zp, 'k-.'); hold on %Plots the point of the true anomaly xa = result_anomaly(1,:); ya = result_anomaly(2,:); za = result_anomaly(3,:); plot3(xa, ya, za, 'ro'); %Make a matrix for all these points X_s = zeros(1,n+1); Y_s = zeros(1,n+1); Z_s = zeros(1,n); Z_s(1,1:(n+1)) = z_south; %Finds the angle for the spacecraft vA = [x_south-xp; y_south-yp; z_south-zp]'; vB = zeros(n,3); angleAB = zeros(n,1); angleSP = zeros(n,1); leftside = 0; rightsideaxis = 0; angleSPw = zeros(n,1); for u=1:(n+1) if zp(1,u)<= - (m_radius+500) if yp(1,u)<= 0 plot3([xp(1,u) xp(1,u)],[yp(1,u) (-1*yp(1,u))], [zp(1,u) zp(1,u)], 'b-.'); vB(u,2) = -2*yp(1,u); else plot3([xp(1,u) xp(1,u)],[yp(1,u) (3*yp(1,u))], [zp(1,u) zp(1,u)], 'b-.'); vB(u,2) = 10000*yp(1,u); end plot3([xp(1,u) X_s(1,u)],[yp(1,u) Y_s(1,u)], [zp(1,u) Z_s(1,u)],'b-.'); angleAB(u,1) = atan2(norm(cross(vA(u,:),vB(u,:))),dot(vA(u,:),vB(u,:)));

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angleAB(u,1) = angleAB(u,1)*180/pi(); angleSP(u,1) = (180- angleAB(u,1))/2; angleSPw(u,1) = (-1/sqrt(1-(angleSP(u,1)/180*pi())^2)); if leftside == 0 leftsideaxis = thetadeg(1,u); leftside = 1; end rightsideaxis = thetadeg(1,u); else vB(u,2) = 0; angleSP(u,1) = 0; angleSPw(u,1)= 0; end end subplot(2,2,4) hold on plot(thetadeg, angleSP); xLabel('Angle (degree)'); yLabel('Angle of Spacecraft (degree)'); %Other values %-------------------------------------------------------------------------------- %Calculates the Time Period Ts = 2*pi()/mu^(1/2)*a^(3/2); %seconds Th = Ts/3600; %hours %Calculates the velocity for each point V = (mu/a).^(1/2)*((1 + 2*e1*cos(theta) +e1^2)/(1-e1^2)).^(1/2); %m/s V_true_a = (mu/a).^(1/2)*((1 + 2*e1*cosd(true_a) +e1^2)/(1-e1^2)).^(1/2); %m/s %Calcuates the energy at each point En = 1/2*V.^2 - mu./r1; %m^2/s^2 %Calcultates the time period from the periapsis to the true anomaly %-------------------------------------------------------------------------------- E = 2*atan(sqrt((1-e1)/(1+e1))*tand(true_a/2)); % calculates the eccentric anomaly MeanA = E - e1*sin(E); %Kepler's Equation t_sp = MeanA/(2*pi())*Ts; %calculates the time in SECONDS %Adjusts the time for the true anomalies above 180 degrees if true_a > 180 t_sp = t_sp + Ts; end t_sp_HOURS = t_sp/3600; %Calcultates the time period from the periapsis to the THETA (matrix) %-------------------------------------------------------------------------------- E_matrix = 2*atan(sqrt((1-e1)/(1+e1))*tan(theta/2)); % calculates the eccentric anomaly MeanA_matrix = E_matrix - e1*sin(E_matrix); %Kepler's Equation t_sp_matrix = MeanA_matrix/(2*pi())*Th; %calculates the time in SECONDS %Adjusts the time for the true anomalies above 180 degrees j = 1; while j <= (n +1) if theta(1, j) > pi() t_sp_matrix (1,j) = t_sp_matrix (1,j) + Th; end

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j = j + 1; end subplot(2,2,3) hold on plot(thetadeg,t_sp_matrix, 'b-.'); plot([180 180], [0 Th], 'r-.'); plot([leftsideaxis leftsideaxis], [0 Th], 'r-.'); plot([rightsideaxis rightsideaxis], [0 Th], 'r-.'); xlabel('Angle (Degree)'); ylabel('Time from Periapsis [hr]'); %Calculates the true anomaly after a given time %-------------------------------------------------------------------------------- t_in = .040736; %Input of time in HOURS MeanAt = 2*pi()*t_in/(Th); %Calculates the mean anomaly given the time input solution_anomaly = kepler_E(e1, MeanAt); %Get the function kepler_E to solve for kepler's law solution_anomaly_degrees = solution_anomaly*180/(pi()); %converts RADS to DEG %MIRROR %-------------------------------------------------------------------------- %Assuming that the mirror is a square (100m^2 or 10mx10m) %then using the simple properties of light rays then the total %power can be calculated. Max_power = 115; %[kW] Power_needed = 200; %[kW] num_sc = 1; %Number of spacecraft in the same orbital at the same position for NOW %%Assume that when the light hits the mirror directly (90 degrees) all the light is transfered to the mirror (max power is achieved). At different angles only a portion of the max power depends on the angle %%Assumptions: at 90 degrees the full power is 100% and after that 0% as %%the spacecraft will be behind the moon. angle = angleSP'; %degrees diffangle = diff(angleSP); %differences addiffangle = zeros(1, n); addiffangle(1,1) = 0; for kk = 1:(n) addiffangle(1,kk+1) = diffangle(kk,1); end %adjusts for the rotating part SPomega = 1; %deg/sec %Initialize the matrices power = zeros(1, (n+1)); percent_power_sc1 = zeros(1, (n+1)); Total_power_percentage = zeros(1, (n+1)); i = 1; while i <= size(angle,2) if angle(1,i)>=0 && angle(1,i) <=90 power(1,i) = Max_power*sind(angle(1, i)); %power received as a function of time percent_power_sc1(1,i) = power(1,i)/Power_needed*100; %percent of power of total needed supplied by one spacecraft Total_power_percentage(1,i) = percent_power_sc1(1, i)*num_sc;

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else Total_power_percentage (i) = 0; end i = i+1; end subplot(2,2,2) hold on plot(thetadeg,Total_power_percentage); xlabel('Angle (degree)'); ylabel('Power (%)'); %There are restrictions to what angle the power is supplied %Looking at the location of the south pole figure plot(t_sp_matrix, addiffangle); %%Output on the Command Window (for now) fprintf('Given: \n'); fprintf('Tin is: %f \n', t_in); fprintf(' Eccentricity(e): %f Semimajor axis(a): %f [km] \n', e1, a); fprintf(' Periapsis Angle: %f Ascending node angle: %f Inclination: %f \n', w, omega,inc); fprintf(' True Anomaly: %f \n', true_a); fprintf('Results: \n'); fprintf(' Period: %f [sec] or %f [hour] \n', Ts, Th); fprintf(' After %f [hr], the spacecraft is %f degrees from the periapsis. \n', t_in, solution_anomaly_degrees); fprintf(' Velocity at this point: %f [km/s] \n', V_true_a); fprintf(' At %f degrees from the periapsis, %f [hr] have passed \n', true_a, t_sp_HOURS); figure plot( t_sp_matrix, thetadeg, 'k-.'); nu = sqrt(mu/a^3); dthetadt = ((sqrt(1+e1)/sqrt(1-e1)*(sec(0.5*E_matrix)).^2)./(sec(0.5*theta).^2))*nu/(1-e1*cos(E_matrix)); difftheta = diff(theta); %differences addiffangle = zeros(1, n); addiffangle(1,1) = 0; for kk = 1:(n) addiffangle(1,kk+1) = diffangle(kk,1); end figure plot(thetadeg, dthetadt, 'ro');

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3.3 Kepler’s Equation (Author: Jesus Ramos)

% ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

function E = kepler_E(e, M)

% ~~~~~~~~~~~~~~~~~~~~~~~~~~~

%

This function uses Newton's method to solve Kepler's

equation E - e*sin(E) = M for the eccentric anomaly,

given the eccentricity and the mean anomaly.

E - eccentric anomaly (radians)

e - eccentricity, passed from the calling program

M - mean anomaly (radians), passed from the calling program

% ----------------------------

Set an error tolerance:

error = 1.e-8;

%...Select a starting value for E:

if M < pi

E = M + e/2;

else

E = M - e/2;

end

%...Iterate on Equation 3.17 until E is determined to within

%...the error tolerance:

ratio = 1;

while abs(ratio) > error

ratio = (E - e*sin(E) - M)/(1 - e*cos(E));

E = E - ratio;

end

end %kepler_E

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3.4 To Calculate Theta and Theta dot as Function of Time. (Author: Kushal Shah)

clear all

clc

a = 5439; %Obtained From Orbit around Moon

e = 0.670927; %Obtained From Orbit around Moon

Mum = 4902.77; %Gravitational Constant for Moon

n= sqrt(Mum/(a^(3))); %Mean Motion of the Orbit around Moon

Inatlizing the Values

Theta_deg = 0.0000001:360/1000:360;

Theta_rad = degtorad(Theta_deg);

E = 2*atan(tan(Theta_rad/2)*(sqrt((1-e)/(1+e)))); %Kepler's Equation to Obtain Eccentric Anomaly

Me = (E - e*sin(E)); %Mean Anomaly.

for i = 1:size(Theta_deg,2)

if Me(1,i) <= 0

Me(1,i) = Me(1,i) + 2*pi();

end

end

t = Me / n; % Solving for time for every theta

thrs = t/3600;

To Get Theta Dot as Function of time

for k = 2:size(Theta_deg,2)

thetadiff = Theta_deg(1,k)-Theta_deg(1,(k-1));

timediff = thrs(1,k)-thrs(1,(k-1));

w3(1,k) = thetadiff/timediff;

end

A = ((sec(E/2)).^(2))/(((sec(Theta_rad/2)).^(2)));

A2 = A./(1-e*cos(E));

A3 = sqrt((1+e)/(1-e))*(n);

A4 = A2 * A3;

Plotting

subplot(2,1,1)

plot (thrs,Theta_deg)%Plotting Theta as Fucntion of time

xlabel('Time (hours)' );

ylabel('Theta (Degrees)');

subplot(2,1,2)

plot (thrs,w3) %Plotting Theta Dot as Fucntion of time

xlabel('Time (hours)' );

ylabel('Theta Dot (Degrees/Hrs)');

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3.5 Calculating Positions of Hohmann Transfer (Author: Kushal Shah)

Parameters Re = 6378.1; %Radius of Earth

MuE = 398600.44; %Specific Gravitational Constant of Earth

Redm = 384400; %Earth to Moon Distance (r2)

LEO Configuration Ralt = 1000; % Altitude of LEO

RLeo = Ralt + Re ; %Radius of LEO respect to ECI (r1)

VLeo = sqrt(MuE/RLeo); %Circular Velocity in LEO

TLeo = (2*pi()*sqrt((RLeo^3)/(MuE))); %Period of LEO

HLeo = sqrt(MuE*RLeo); %Momentum in LEO

Plots the Earth in [x y z] = sphere();

sphere = surf(Re*x, Re*y, Re*z);

set(sphere, 'FaceColor','Blue', 'FaceAlpha',1);

daspect([1 1 1]);

xlabel('X axis');

ylabel('Y axis');

zlabel('Z axis');

hold on

Values for the orbital elements k = input('moon(1) , s/c (2): ');

if k == 1

a= ((Redm)); %Semi-Major Axis of Hohmann for LEO to MOON

else

a= (0.5*(Redm+RLeo));

end

if (a == Redm)

e1 = 0;

else

e1 = (1 -(RLeo/a));% eccentricity of Hohmann for LEO to MOON

end

p1 = a*(1-e1^2);

knew= input('enter true anamaly: ');

%Fixes the angle (INPUTS)

true_adeg = knew; %

inc = 5.26; % inclination angle

omega = 226.75; %Longitude of ascending node angle

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if k == 1

w= 0; %Semi-Major Axis of Hohmann for LEO to MOON

else

wnew = input('enter the w deviation: ');

w= wnew;

end; %argument of periapsis angle

%Angle from 0 to 2Pi

theta = (0: 2*pi()/1000: 2*pi());

Rotation Matrices R_omega = [cosd(-omega) sind(-omega) 0; -sind(-omega) cosd(-omega) 0; 0 0 1]; %about the 3rd axis

R_inc = [1 0 0; 0 cosd(-inc) sind(-inc); 0 -sind(-inc) cosd(-inc)]; %about the 1st axis

R_w = [cosd(-w) sind(-w) 0; -sind(-w) cosd(-w) 0; 0 0 1]; %about the 3rd axis

Ploting Orbit r1 = p1./(1 + e1*cos(theta));

z0 = zeros(size(theta));

%%Converts from polar to cartesian coordinates

[X1,Y1, Z1] = pol2cart(theta, r1, z0);

% Result if after the rotation of all three angles

result = R_omega*R_inc*R_w*[X1; Y1; Z1];

% Gets each value (x, y, z) and plots the orbit

xp = result(1,:);

yp = result(2,:);

zp = result(3,:);

plot3(xp, yp, zp, 'k-.');

hold on

Given True anomaly S/C 1

true_arad = degtorad(true_adeg); %true anomaly

r_ta = p1./(1 + e1*cos(true_arad));

[Xa, Ya, Za] = pol2cart(true_arad, r_ta, 0);

Mean_eccen = 2*atan(tan(true_arad/2))/(sqrt((1+e1)/(1-e1)));

Mean_eccendeg = radtodeg(Mean_eccen);

%Result after the rotation for the true anomaly

result_anomaly = R_omega*R_inc*R_w*[Xa; Ya; Za];

%Plots the point of the true anomaly

xa = result_anomaly(1,:)

ya = result_anomaly(2,:)

za = result_anomaly(3,:)

plot3(xa, ya, za, 'ko');

Getting Time and Angles: Ts = (2*pi()/MuE^(1/2))*a^(3/2); %seconds

Th = (Ts/3600)*.5;

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3.6 Calculating ΔV for Hohmann, Phase Transfer and Plane Change (Author: Kushal

Shah)

clc

clear all

Parameters

Re = 6378.1; %Radius of Earth

MuE = 398600.44;

MuM = 4902.77; %Specific Gravitational Constant of Earth

Redm = 384400; %Earth to Moon Distance (r2)

LEO Configuration

Ralt = 1000 ; % Altitude of LEO

e = 0;

RLeo = Re + Ralt ; %Radius of LEO respect to ECI (r1)

VLeo = sqrt(MuE/RLeo); %Circular Velocity in LEO

TLeo = (2*pi()*sqrt((RLeo^3)/(MuE))); %Period of LEO

HLeo = sqrt(MuE*RLeo); %Momentum in LEO

Phasing the S/C in LEO

Angleofphase = 17;

% Phase Change Manuver

true_arad = degtorad(Angleofphase); %true anomaly

Mean_eccen = 2*atan(tan(true_arad/2))/(sqrt((1+e)/(1-e)));

Mean_eccendeg = radtodeg(Mean_eccen);

tab = (TLeo / (2* pi()))*Mean_eccen;

Tneeded = TLeo -tab;

aneeded = ((sqrt(MuE)*Tneeded)/(2*pi()))^(2/3);

rc = 2*aneeded - RLeo;

h2 = sqrt(2*MuE)*sqrt((RLeo*rc)/(RLeo+rc));

V2 = h2 / RLeo;

DelVbeg = V2 - VLeo;

DelVend = VLeo - V2;

phasedelt = abs(DelVbeg)+abs(DelVend)

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Hohmann Transfer

aLtm = ((RLeo + Redm)* 0.5); %Semi-Major Axis of Hohmann for LEO to MOON

eLtm = (1 -(RLeo/aLtm));% eccentricity of Hohmann for LEO to MOON

delv1 = (sqrt(MuE/RLeo)*((sqrt((2*Redm)/(RLeo+Redm))-1))); %Delta v of Hohmann for LEO to Moon

delv2 = (sqrt(MuE/Redm)*(1-(sqrt((2*RLeo)/(RLeo+Redm))))); %Delta v of Hohmann for LEO to MOON

Hohdelt = delv1+delv2;

Vper = VLeo + delv1;

Vper2 = (sqrt(MuE/aLtm)*(sqrt((1+eLtm)/(1-eLtm)))); %perprisas velocity of Hohmann for LEO to MOON

Vapp = (sqrt(MuE/aLtm)*(sqrt((1-eLtm)/(1+eLtm)))); %apoapsis velocity of Hohmann for LEO to MOON

Vmoon = sqrt(MuE/Redm) ;

Vmoodes= sqrt(MuE/Redm);

delv22 = Vmoon+Vmoodes - Vapp;

THohm = (2*pi()*sqrt((aLtm^3)/(MuE)))/(3600*2);

Calculating Mass Needed from total delta V

VTOTOAL = phasedelt+Hohdelt+0.0933 %(0.0933 Calculated from Inclanation Phase Change)

Ms = 500;

Mp = 100;

ISP = 250;

Ve = 9.81*ISP/1000;

ap = exp(VTOTOAL/Ve);

Mf = (Ms+Mp)*(ap-1); %Mass Calculated from Total DeltaV

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3.7 Calculating initial and final positions (Author: Kushal)

clc

clear all

MuE = 398600.44;

n = sqrt((MuE)/((1000+6378.1)^3));

t= (.438) + 2*(105.1179/60)+22.79; % time of Arrival

Thetainitital= 200.5439- radtodeg((n)*(t*3600)) % solving for Initial positions

k1 = -Thetainitital / 360;

if k1 >= 1

k2 = (int16(fix(k1)))*360;

k3 = Thetainitital + k2;

k4 = k3 + 360;

Thetainitital1 = k4

end

if k1 <= 1

if Thetainitital <=0

Thetainitital2 = Thetainitital + 360

end

end

tlaunch = t

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X. Work Cited(Team) [1] Jennifer Chu. “Researchers find evidence of ice content at the moon’s south pole.” MIT News

Office. 20 June 2012. Web. <http://web.mit.edu/newsoffice/2012/shackletons-crater-0620.html>

[2] David Darling. “Moon Base.” <http://www.daviddarling.info/encyclopedia/M/Moon_base.html>

[3] Giorgini, Jon (author of the code). "HORIZONS Web-Interface." HORIZONS Web-Interface.

Solar System Dynamics Group, Horizons On-Line Ephemeris System, n.d. Web. 03 Mar. 2013

<http://ssd.jpl.nasa.gov/horizons.cgi#results>.

[4] Curtis D., Howard. Orbital Mechanics for Engineering Students. 2nd. Oxford, UK: Butterworth -

Heinemann, Print.

[5] Yvette Smith. Space Missions. http://www.nasa.gov/missions