computation of planetaryorbits

8/10/2019 Computation of PlanetaryOrbits

1/9

Computation of Planetary OrbitsAuthor(s): Donald A. Teets and Karen WhiteheadSource: The College Mathematics Journal, Vol. 29, No. 5 (Nov., 1998), pp. 397-404Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2687254.

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2/9

Computation

of

Planetary

Orbits

Donald

A.

Teets and

Karen

Whitehead

Donald

Teets

([email protected])

received

his

B.A. from the

University

of

Colorado,

his M.S.

from

Colorado State University, nd his Doctor of Arts from Idaho

State

University.

He has

taught

at

the

South Dakota School

of

Mines and

Technology

since

1988 and

has been chair of

the

Department

of

Mathematics

and

Computer

Science

since 1997. When he is not

doing

mathematics,

he

enjoys

backpacking, cross-country

skiing,

and rock

climbing.

Karen

Whitehead

([email protected])

received her

B.A.

and

Ph.D.

from the

University

of Minnesota.

She

joined

the

faculty

at the

South

Dakota

School of

Mines and

Technology

in

1981,

serving

as

department

head and

college

dean

before

taking

on her current duties

as Vice

President forAcademic Affairs. Her major avocation is

music:

she is

a substitute

church

organist

and

has

sung

in

student

choral

groups

on

campus

for the

past

14

years.

It is uncommon to

find in the

histoiy

of science and

mathematics

great problems

whose solutions are accessible

to first-

or

second-year undergraduates.

The

descrip?

tion

and

computation

of

planetary

orbits

provides

a wonderful

opportunity

to

study

just

such a

problem.

Although

techniques

for

computing

orbits

are

certainly

well

known

and

can be found

in

many

textbooks on

celestial

mechanics

[2, 5-8],

we of?

fer

here

a concise introduction

to the

subject

intended for

students

with

only

a

strong

background

in the calculus and

analytic geometry.

No

background

in

astronomy

is

assumed. The mathematical

tools encountered in this

problem

include

vector cross

products,

dot

products, equations

of

lines and

planes

in

space,

basic

geometric

and

trigonometric

relationships,

the

trapezoidal

rule,

the

polar

coordinate

equation

of an

ellipse,

area in

polar

coordinates,

and

a few other ideas that students

should

find

familiar. What

a

treat

to

find an

application involving

such a

wide

variety

of

topics

from a standard

course in calculus

and

analytic geometry

Our mathematical

presentation

has

two

limitations that we are aware of.

First,

our

goal

is

to

present

a method

that is

accessible

to students with a modest

background,

rather than a method

for

experts

that will

produce

highly

accurate results.

The second

flaw

is more

fundamental:

We

assume that

the

planet's position

with

respect

to

the

sun

is

completely

known at two

different

times. In

practice,

the

problem

of

obtaining

two

such

position

vectors

from earth-based observations

is substantial?it

was

Gauss's

solution

to this

problem

(and

the

subsequent

orbit

computation

for the

minor

planet

Ceres)

that "made Gauss a

European celebrity"

in

1802

[1],

We state

the

problem

as follows:

Given two vectors

in

M3

describing

the

positions

ofa

planet

with

respect

to the sun

at

two

different

times,

compute

parameters necessary

to

determine

completely

the

planet's

position

at

any

specified

time.

We

accept

as

given Kepler's

three

laws of

planetary

motion:

a

planet's

orbit

is an

ellipse with the sun at one focus; the planet's speed changes along the orbit in such

VOL.

29,

NO.

5,

NOVEMBER

1998

397

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3/9

a

way

that

the radial vector

from

the sun to the

planet sweeps

out

area at a constant

rate;

and

the ratio of the

square

of the

orbital

period

to the cube of

the

length

of the

semimajor

axis is the same

for

all

planets

in our solar

system.

Accessible

derivations

of

these laws

can be

found in

[6].

Our

problem

is to show how the

shape

of the

elliptical

orbit,

its

orientation

in

space,

and the motion

of

the

planet along

the

orbit

can all be determined from the two

given

radial vectors.

Basic

terminology.

To

develop

the basic

terminology

of

planetary

orbits,

our first

task is

to

establish

a

coordinate

system

in

space,

as illustrated

in

Figure

1.

/*

First

day

3L

of

spring

First

day^\

/sun /

of summer

\

/

/

Figure

1

We shall use

a

rectangular

coordinate

system

with the

sun

at

the

origin

and with the

XF-plane

chosen to

be

the

plane

ofthe earth's

orbit,

the

so-called

ecliptic plane.

The

line

through

the earth and sun on the first

day

of

spring

(in

the northern

hemisphere)

is chosen

as the

X-axis,

and

the

F-axis

is,

of

course,

chosen

perpendicular

to the

X-axis. These

axes are

directed

so that the

sun,

as viewed from

earth,

appears

in

the

positive

X

direction on the first

day

of

spring

and

in

the

positive

Y

direction

as

summer

begins.

The

Z-axis is

then

chosen

to

form

a

right-handed

coordinate

system.

The

angle

i

between the

positive

Z-axis and

the

vector

n normal to

the

planet's

orbital

plane (Figure

2)

is

called

the inclination

of the orbit.

(Angles

are measured

in

degrees

unless

noted

otherwise.)

We shall assume

that

0?

computation of planetaryorbits

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