STEREO Orbital Kinematics
Author :
Dr. Anand Balaraman,
Department of Physics, Georgia Southern University.
Email: [email protected]
Abstract: This lab involves using the STEREO Orbit Tool to find the angular positions
of the STEREO space crafts on a start date and use the rotational kinematic equations to
predict the evolution of these quantities with time. Using the fundamental equations the
students are asked to predict when the phase difference between the satellites reach certain
mission milestones.
Audience: Appropriate for the freshmen level introductory physics courses on Newtonian
mechanics.
Educational Goals : The primary goal of this module is to assist the students in understanding the basic concepts of rotational kinematics and help them acquire problem solving
skills using the rotational equation of motion. The secondary goal is to introduce them to
the interesting kinematics of the STEREO orbiter mission, a very important mission for heliophysics. The primary goal is achieved by letting the students apply rotational equations
of motion to predict the positions of the two STEREO space crafts relative to each other
and to Earth.
Module Type : In class activity guided by the instructor followed by a group lab activity.
Total duration would be 75 - 90 minutes.
Module Outline :
1. Read the introduction section to understand the basic idea behind the STEREO mission.
2. Using the STEREO Orbit Tool, note down the angular positions of the STEREO
satellites relative to Earth on some ”start date” (March 01, 2007).
3. Taking these as the initial values, calculate their subsequent evolution by employing a
simplified model of circular orbits for the STEREO satellites and Earth.
4. Use the rotational equations of motion to predict the year & month when the phase
difference between the satellites reach certain milestones.
Resources Needed : Requires computers connected to internet. (specify browser configuration requirements)
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STEREO Orbital Kinematics
Objectives
Objective of this lab is to apply the concepts learned in rotational kinematics to study the
orbital kinematics of the STEREO space crafts. The students will:
• use the STEREO Orbiter Tool to find the angular positions of the two STEREO
space crafts, relative to Earth,
• model their motion as uniform circular motion and use the rotational equation of
motion to calculate the angular positions of the two STEREO space crafts, and
• predict when the phase difference between them reach certain milestones.
Background Information
STEREO (Solar TErestrial RElations Observatory) is a solar orbiter mission with the goal
of observing the Sun from two different directions. The mission employs two space crafts,
STEREO-A and STEREO-B, in heliocentric orbits. STEREO-A goes around the Sun in an
orbit that lies just inside Earth’s orbit with an angular speed higher than Earth’s angular
speed. STEREO-B goes around the Sun in an orbit that lies just outside Earth’s orbit
with an angular speed lower than Earth’s angular speed. Because of the differences in their
angular speeds, the space crafts appear to drift away from Earth. STEREO-A appear to
be going ahead of Earth and STEREO-B appear to be going behind the Earth in its orbit.
The mission was launched on October 26, 2006. On December 15, 2006 STEREO-A was
transferred from a highly eccentric geocentric orbit to a nearly circular heliocentric orbit
with an eccentricity of just 0.006. On January 21, 2007 STEREO-B was transferred to a
an elliptical heliocentric orbit with an eccentricity of 0.042. For comparison the eccentricity
of Earth’s orbit is 0.017. So compared to Earth, STEREO-A has a more circular orbit
and STEREO-B has a more elliptical orbit. For this lab activity we will ignore the orbital
eccentricities and model the motion of Earth and the two space crafts as Uniform Circular
Motion.
The STEREO Orbit Tool
• Click on the following URL to go to NASA’s STEREO Science Center web page which
archives STEREO mission data and functions as a portal for science education and
public outreach activities.
http://stereo.gsfc.nasa.gov/
• Explore the page and click on the link that says Where Is STEREO?. Scroll all the
way down and locate the link to the STEREO Orbiter Tool If you cannot locate it
then just click on the URL given below. It will take you to the Orbiter Tool directly.
http://stereo-ssc.nascom.nasa.gov/cgi-bin/make_where_gif
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• The STEREO Orbiter Tool allows the user to select a date and time from a drop
down list and calculates the positions of the inner planets and the two STEREO space
crafts. Besides the inner planets and the STEREO space crafts, there are options
to make the tool calculate the positions of other space crafts in the vicinity. The
calculated positions of the inner planets and the space crafts are shown as a plot. The
plot shows a cartesian grid in which Sun is located at the coordinate origin. The grid
is actually rotating with the Earth so that Earth is always located at a distance of
1.0 AU on the Y-axis. Below the plot some important numerical data, regarding the
positions of Earth and the two STEREO space crafts, are also provided. Heliocentric
distance is the straight line distance between the Sun and the objects. The straight
line distance between the two space crafts and the Earth is given in terms of the light
travel time to Earth. More importantly the angular separations between the two
space crafts and the Earth are given in the last two rows (Separation angle with
Earth & Separation angle A with B)
Activities
(1) Ignoring the orbital eccentricities, model the motion of Earth and the two STEREO
spacecrafts as Uniform Circular Motion (constant angular speed). The orbital time
period of Earth, STEREO-A and STEREO-B are, respectively 365.2 days, 344.5 days
& 389.0 days. From the orbital time periods calculate the angular speeds of these
objects and Answer Comprehension Question 1.
(2) Though the STEREO mission was launched in October 2006, it took about 3 months
to transfer them from geocentric orbits to heliocentric orbits. After a few orbital
adjustments the space crafts settled in their orbits by February 2007. Allowing little
more time, let us take March 01, 2007; 00:00 UT as the starting point from which we
will start the calculations. Let us define the X-axis as the line connecting Sun and
Earth on that date at that time. So the initial angular position of Earth is zero by
definition. The situation is illustrated (Figure 1). Use the date/time drop down list
under the plot region to set the date and time to March 01, 2007; 00:00 UT and note
down the angular positions of STEREO-A and STEREO-B. These are the initial values
of their angular positions. Use the rotational equations of motion to write down their
angular position at time ’t’ after this point. Answer Comprehension Questions 2
& 3.
(3) Because of the differences in the angular speeds, the two space crafts appear to drift
away from Earth. STEREO-A appears to drift ahead of Earth in its orbit while
STEREO-B appears to drift behind the Earth. From the equations of motion calculate
the rate at which the two space crafts drift away from Earth. Answer Comprehension Question 4.
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Figure 1: Positions of Earth, STEREO-A & STEREO-B on Mar 01, 2007.
(4) The angular separation between the space crafts too increase with time. As they drift
away from Earth the space crafts pass through interesting phases marking the different
mission milestones. These milestones are discussed below in detail. Use the equations
of motion and estimate the approximate time (year and month) these milestones will
be reached. The STEREO Orbiter Tool helps you to find the exact time these
events occurred.
– Quadrature: When the angular separation between the crafts reach 90◦ the coronal mass ejections seen from one space craft will pass through the other craft if the
ejection was directed that way. Estimate the approximate time (year/month/day)
when the angular separation between the two space crafts approach 90◦ ? Then
use the Orbiter Tool and find the exact date this happens. Answer Comprehension Question 5.
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– Lagrange Points Approach: The Earth-Sun Lagrange points, L4 and L5 , lie
60◦ ahead and behind the Earth along its orbit. As the space crafts pass through
these points they can look for trapped trojan asteroids. Because of the differences in their rate of drift away from the Earth, these space crafts are expected
to arrive at the corresponding Lagrange points on different dates. Estimate the
approximate time (year/month/day) when the two space crafts reach these Lagrange points. Do these calculations for STEREO-A and STEREO-B separately.
Then use the Orbiter Tool to find the exact date these events happen. Answer
Comprehension Question 6
– Full View Of Sun: When the angular separation between the space crafts reach
180◦ we get a view of the entire solar surface for the first time. Estimate the
approximate time (year/month/day) when the angular separation between the
two space crafts approach 180◦ ? Then use the Orbiter Tool and find the exact
date this happens. Answer Comprehension Question 7.
– Behind The Sun: When the angular separation between Earth and the space
crafts approach 180◦ they will be eclipsed by the Sun. Again, this event is expected
to happen at different times for the two space crafts. Estimate the approximate
time (year/month/day) these events happen. Then use the Orbiter Tool to find
the exact dates these events occur. Answer Comprehension Question 8.
(5) Look at the data in Table 1 and compare the Uniform Motion Model based estimated
dates for these events with the exact dates found using the STEREO Orbiter Tool.
Answer Comprehension Question 9.
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Comprehension Questions
Question 1: From the given orbital time periods, calculate the angular speeds of
Earth and the two STEREO space crafts? Do not forget to convert the unit to ◦ /year.
[1 year ≡ 365 days]
Earth (TE = 365.2 days) :
ωE =
◦
/year.
STEREO-A (TA = 344.5 days) :
ωA =
◦
/year.
STEREO-B (TB = 389.0 days) :
ωB =
◦
/year.
Question 2: Note down the initial values of the angular positions of STEREO-A and
STEREO-B.
0.0◦
Earth
:
θE0 ≡
STEREO-A
:
θA0 =
◦
.
STEREO-B :
θB0 =
◦
.
(Mar 01, 2007; 00:00 UT)
Question 3: Write down the equations for the angular positions of Earth and the two
space crafts at time ’t’ after the start date/time?
Earth
:
θE (t) =
STEREO-A :
θA (t) =
.
STEREO-B :
θB (t) =
.
Question 4: From the equations of motion, determine the rate at which the two space
crafts drift away from Earth?
Earth - STEREO-A
:
∆θAE
=
∆t
◦
/year.
Earth - STEREO-B
:
∆θBE
=
∆t
◦
/year.
θAE = θA − θE
:
angular separation between Earth and STEREO-A,
θBE = θB − θE
:
angular separation between Earth and STEREO-B,
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Question 5: When do the two space crafts reach the Quadrature configuration? Use
the first row of Table 1 to enter your answers.
Question 6: When does STEREO-A approach the Lagrange points L4 (θAE = +60◦ )?
When does STEREO-B approach the Lagrange point L5 (θBE = −60◦ )? Use the second
row of Table 1 to enter your answers.
Question 7: When do the two space crafts get to the position where they are right
opposite of each other, enabling us to get a full view of the SunθAB = 180◦ ? Use the
third row of Table 1 to enter your answers.
Question 8: When do STEREO-A and STEREO-B go behind the Sun? Use the
fourth row of Table 1 to enter your answers.
Question 9 How big are the differences between the model predictions and the exact
dates? What is causing this discrepancy?
References
[1] NASA - Solar Physics Group’s Webpage : http://solarscience.msfc.nasa.gov/
[2] UCLA - Space Physics Center : http://www-ssc.igpp.ucla.edu/forms/stereo/
[3] SECCI - STEREO Space Weather Group : http://secchi.nrl.navy.mil/spwx/
index.php
[4] NASA - STEREO Mission Page : http://www.nasa.gov/mission_pages/stereo/
main/index.html#.Uuwf8j1dUuc
[5] NASA - STEREO Science Center : http://stereo-ssc.nascom.nasa.gov/where.
shtml
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Table 1: STEREO Mission Milestones
Question
Mission Milestones
Q5
Quadrature
(θAB = 90◦ )
Event Dates
Model Calculation
Orbiter Tool
[Month/Year/Day]
[Month/Year/Day]
Lagrange Points Approach
Q6
STEREO-A (θAE = 60◦ )
STEREO-B (θBE = 60◦ )
Q7
Full View Of The Sun
(θAB = 180◦ )
Behind The Sun
Q8
STEREO-A(θAE = 180◦ )
STEREO-B(θBE = 180◦ )
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