Exercise 1 Orbital Mechanics - Lab-Volt · orbital mechanics 7kh vsdfh vhjphqw ri d vdwhoolwh v\vwhp frqvlvwv ri rqh ru pruh vdwhoolwhv lq vsdfh lqfoxglqj erwk dfwlyh dqg vsduh vdwhoolwhv

$Page 1: Exercise 1 Orbital Mechanics - Lab-Volt · orbital mechanics 7kh vsdfh vhjphqw ri d vdwhoolwh v\vwhp frqvlvwv ri rqh ru pruh vdwhoolwhv lq vsdfh lqfoxglqj erwk dfwlyh dqg vsduh vdwhoolwhv$
© Festo Didactic 87768-00 11

Simplifying assumptions. Kepler’s laws as they apply to satellites. Newton’s laws of motion. Vectors and scalars. Newton’s law of universal gravitation.

Inertial and non-inertial reference frames. Coordinate systems. The earth-centered inertial (ECI) coordinate system. Angles in the ECI coordinate system. The earth-fixed Greenwich (EFG) coordinate system. Geodetic and geocentric latitude. Sidereal time and solar time.

Orbital state vectors. Conservation of angular momentum and mechanical energy. Keplerian orbital elements. Position of the satellite in the orbit. Perigee and periapsis. Anomalies. Radius and altitude. Other orbital elements.

By period. By altitude. By inclination. By eccentricity.

Orbital mechanics

Orbital Mechanics

Exercise 1

EXERCISE OBJECTIVE

DISCUSSION OUTLINE

DISCUSSION

Exercise 1 – Orbital Mechanics Discussion

12 © Festo Didactic 87768-00

Philosophiæ Naturalis Principia MathematicaPrincipia

Physical laws

Simplifying assumptions

Kepler’s laws as they apply to satellites





Newton’s laws of motion

same speed same direction

magnitudedirection

F a



tt

t



Newton’s law of universal gravitation

The ellipse



mean distance



Reference frames and coordinate systems



Inertial and non-inertial reference frames



not acceleratingphysical frame of reference

frame of reference refer-

ence frame frame



Coordinate systems

a In orbital mechanics, several different coordinate systems are commonly used. As a coordinate system is always associated with a specific frame of reference, the term “frame of reference” often refers to both an observational frame of reference and to an attached coordinate system.

origin



or

or



a In the geographical coordinate system, the symbol is frequently used for longitude, the signed angular distance of a location measured eastward from the prime meridian, and the symbol is often used for latitude, the signed angular distance of a location north or south of the equator.

The earth-centered inertial (ECI) coordinate system



vernal equinox

direction



equatorial coordinate system

right ascension declination



Sidereal time

The earth-fixed Greenwich (EFG) coordinate system



a All of these angles are measured from the center of the earth, which is in the equatorial plane.

sidereal time (ST)Sidereal time and solar time

longitude geocentric latitude





topocentric toposhorizontal





Defining an orbit

Orbital state vectors

r v

epoch

time t .

r v



r v

r

v



specific mechanical energy



Keplerian orbital elements

anomaly



a e

i

r

v



a Strictly speaking, inclination is never negative, since it is measured at the ascending node. In some cases, however, the concept of negative inclination is useful. For example, the inclination of geostationary satellites tends to drift slowly towards more positive values. To maintain the inclination near zero, thrusters are fired periodically to reduce the inclination to a slightly negative value, after which it begins to drift positive again.

In the LVSAT Orbit Simulator, inclination can be set between ±180°.

apsides

perigeeapogee perihelion aphelion

perigee periapsis

argument of periapsis

longitude of the ascend-

ing node



Anomalies

a a E

ae

a

E



area

time

:







r

r

Orbit classifications

By period

a multiple

equal



By altitude



h

By inclination

direct prograde orbit

retrograde orbit

By eccentricity



Subsatellite point and ground track

Exercise 1 – Orbital Mechanics Procedure Outline


a The first section of the following Procedure is designed to familiarize you with the LVSAT Orbit Simulator. In order to save time, it is suggested that you read the information presented in yellow boxes in this familiarization section before beginning your first laboratory period with the Orbit Simulator.

General Settings. Settings tables. Time in the Orbit Simulator. Satellite Editor.

Orbital and apparent paths.

a The procedure of this exercise is divided into sections which are independent. As the procedure is fairly long, you may wish to do it in two lab periods.

Familiarization with the LVSAT Orbit Simulator

This section will allow you to become familiar with various aspects of the Orbit Simulator. Sidebars in this section present information on windows, views, the active satellite, settings tables, controlling time, and using the Satellite Editor.

Windows

views

PROCEDURE OUTLINE

PROCEDURE

Exercise 1 – Orbital Mechanics Procedure


b



Views

a The Orbit view shows the relative size of the earth as a blue circle. Increasing

the semi-major axis of the orbit may only reduce the size of the blue circle, and

vice-versa.

b To pan in this view, click and drag with the mouse. To zoom in or out, roll the

mouse wheel or press the Page Up or Page Down keys.

b To view the earth from a different angle, click in the view and drag with the

mouse. This changes the camera’s angular position in the current Camera

Frame (ECI and EFG Camera Frames only). To view the earth closer up or

further away, roll the mouse wheel or press the Page Up or Page Down keys.

This changes the altitude of the camera. For fine adjustments, press and hold

the Ctrl key first.

a A light source is used in the view to produce a three-dimensional effect.

This light source has no relation to the sun; the sun is not modeled in the Orbit

Simulator.



Active satellite

b You can click on a satellite in the 3D view to make it the active satellite.

Active Satellite

General Settings



Time in the Orbit Simulator

Controlling time

t

a Real solar time is measured by the apparent movement of the sun. The length

of a solar day is 24:00:00. Since the sun is not represented in the Orbit

Simulator, the displayed Solar Time only represents the number of solar hours

elapsed since time was started or reset.

t0 M

Mean Anomaly at Epoch M0

sidereal time



Run

Pause

Reset

a Changing the Speed of time, or pausing time, makes the animation run faster

or slower, or stops it altogether. However, this does not change the Position

Rates displayed in the Information window. For example, a geostationary

satellite has a speed of approximately 3.07 km/s regardless of the current

Speed of time in the animation.

b When time is paused, you can click in the Time window and roll the mouse

wheel to advance or recede time by small increments that depend on the

magnitude of the current Speed setting. For fine control of time when using

the mouse wheel, set the Speed to a low value.

bb





Satellite Editor

Using the Satellite Editor

SatellitesEdit

Add

Remove



Reference frames and coordinate systems

In this section, you will show the different axes available in the Orbit Simulator. You will use the Camera Frame setting to change the reference frame used to generate the image presented to the user.

a Initial conditions: All satellites have been removed.



Camera Frame

Camera Frame

Camera Frame

Camera Frame

Camera Frame



b



a In the Satellite Camera Frame, you cannot change the angular position of the camera. You can, however, move the camera above or below the satellite.

Types of orbits

In this section, you will load a file containing satellites with different types of orbits. You will observe how the orbital elements determine the type of orbit and its characteristics.



File Load SatellitesYes

a The pre-configured satellite files are typically installed in the folder: C:\Program Files\Lab-Volt\LVSat\Orbit Simulator\Satellite Files.

It is suggested that you store user-created satellite files in a different folder.

a By default, a “P” identifies the perigee of each orbit. However, when the eccentricity , the orbit is circular and there is no perigee. In this case, the “P” shows where the perigee would be, according to the current Argument of Perigee setting, if the eccentricity were to become non-zero.





File Save Satellites As

b



Apparent paths

In this section, you will observe how the paths of various satellites actually appear to an observer using two different camera frames.

b



b



b



b

Orbit shape and size

In this section, you will observe how the orbital elements of a satellite determine the shape and size of its orbit.



b

b



b



State vectors and their components

In this section, you will observe the state vectors of a satellite. You will note the values of the components of these vectors, expressed in spherical coordinates, as displayed in the Information window. For different types of orbits, you will note which components are fixed and which are constantly changing.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0

200

400

600

800

1000

1200

1400

1600

10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000



b





a As only the principles are important, do not include units in your calculations. The rounding off of values displayed in the Information window may introduce a small error in your calculations.

Orientation of the orbit and position in orbit

In this section, you will observe how the orbital elements of a satellite determine the orientation of the orbit in space and the position of the satellite in its orbit.

b



a For a perfectly equatorial orbit (inclination = 0 ), the RAAN is undefined. Since the orbit is in the plane of the equator, there is no ascending node. In the Orbit Simulator, however, the RAAN is always defined as the angle at which the ascending node would appear if the inclination became non-zero.

a For each orbit, a “P” in the 3D view identifies the perigee. For a perfectly circular orbit (eccentricity = 0), there is no perigee and the Argument of Perigee as well as all of the anomalies are undefined. In the Orbit Simulator, however, the “P” shows where the perigee would be located, according to the current Argument of Perigee setting, if the eccentricity became non-zero. The anomalies are defined as the angles measured from this point.



Subsatellite point and ground track

In this section, you will observe the subsatellite point and ground track for several different types of orbits. (As the procedure of this exercise is rather long, this final section is optional.)

b

b



b

b

Tools Options



b

Exercise 1 – Orbital Mechanics Conclusion


tp

CONCLUSION

REVIEW QUESTIONS

Exercise 1 – Orbital Mechanics Review Questions


equatorial orbit polar orbit prograde orbit retrograde orbit

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Exercise 1 Orbital Mechanics - Lab-Volt · orbital mechanics 7kh vsdfh vhjphqw ri d vdwhoolwh v\vwhp frqvlvwv ri rqh ru pruh vdwhoolwhv lq vsdfh lqfoxglqj erwk dfwlyh dqg vsduh vdwhoolwhv