# The Physics, Astronomy and Mathematics of the Solar System

## Mathematics at the Frontier of Astronomy

Your feedback is important to us!

After viewing our curriculum units, please take a few minutes to help us understand how the units, which were created by public school teachers, may be useful to others.

## Measurements in Space

### Gravity

Gravitational attraction is a fundamental property of matter. The gravitational force is extremely weak. It is the weakest of the four known forces in nature. The other forces are the strong and weak force, nuclear forces, and the electromagnetic force. Gravitational force is of importance when one of the bodies has a very large mass, as large as a planet or a star, or any other large celestial body. Sir Isaac Newton described mathematically this universal force of gravity. He found that gravity extended above the domain of Earth to other bodies.

The measurement for gravity '
*
G
*
' was determined by a British scientist Henry Cavendish in 1797. For his experiment two small spherical masses of the same weigh m in kilograms were attached to the ends of al light rod. This system was suspended by a fine wire so that the system is balanced. Two larger masses M were placed at the opposite ends of the smaller masses. The gravitational attraction between the masses twisted the wire. The elastic property of the wire was known, so it was possible to measure the force of attraction between the masses m and M. Substituting the found measurement in the equation
*
F
*
_{
g r a v i t y
}
= GmM/r
^{
2
}
. The value 6.67 × 10
^{
11
}
N.m
^{
2
}
/kg
^{
2
}
is used for
*
G
*
.

Newton's second law for freely falling bodies of mass M near the Earth's surface and the value of
*
G
*
and GmM/R^2 = mg, was used to calculate the mass of the Earth. Where
*
M
*
is the mass of the Earth and
*
R
*
the radius of the Earth the calculations gives [(6.67 * 10^-11 * Nm^2 / kg^2)*M] / (6.38 x 10^6 *m)^2 = 9.8m/s^2. This gives
*
M = 5.98 × 10
^{
24
}
*
kg for the mass of the Earth.

### Cosmic Distances

The distance of a star from sun is very large. The approximate distance of the closest star to the Earth is almost 250,000 times the distance of the Earth to the Sun. Because of these great distances no direct measurement of these distances can be applied. To measure stellar distance astronomers have used geometrical method to determine these distances. This process is called parallax measurement
^{
15
}
.

Parallax is defined as the angular or apparent change in the position of an object due to the change in the position of the observer. A distant star will appear to change its position as the Earth moves on its orbit around the Sun. The angle between the two lines of sight is called the parallax angle. The size of the parallax angle is dependent on the distance between the observation point and the distance being observed. The parallax angle of a star determines the distance to the star. The further away the star is the smaller the parallax angle.

It is important to have consistency when comparing the parallax angle of different stars thus determining their distances. Astronomers agree to use 1 AU
^{
16
}
as the same base line for the two observations that will be used for measuring the parallax. Thus the parallax of a star is the parallax angle that will be measured from two points separated by exactly one astronomical unit and with the line between the points perpendicular to the direction of the star. The value determined is called the trigonometric parallax.

To measure a star's parallax astronomers measure a star's distance using two points of view.

- 1. When the Earth is at the point P
_{ 1 }in its orbit, set the telescope to receive light from the star to be measured and the distant sky beyond. The star should be in the center of the view.- 2. Six months later when the Earth is at point P
_{ 2 }with the telescope to receive light from the same star under the same conditions, turn the telescope through twice the angle of the previous setting to place the star in center the of view.- 3. Since the distance from the Sun to the Earth is known the measurement of the angle generates a right triangle. Trigonometry can be used to measure the side of the triangle that corresponds to the distance from the Sun to the star.

The direction from the Earth to the star changes as the planet orbits the sun, and the nearby star appears to move back and forth against the background of more distant stars. The parallax of a star apparent position shifts as the Earth moves from one side of its orbit to the other. The larger the parallax "
*
p
*
" the smaller the distance '
*
d
*
' to the star.

*
Let P
_{
1
}
*
= the first observation

*
Let P
_{
2
}
*
= the second observation (six months later)

*
Let p
*
= the parallax (angle between the line of sight and the perpendicular line from the sun)

*
Let d
*
= the distance to the stars.

The distance d is measured in parsecs. The unit of measure called the parsec written as pc is defined to be the distance for which the parallax is exactly one second.
*
If p = 1
*
arcsecond, by definition
*
d = 1 pc
*
.

The following relationship can be written between p (the parallax) and d the distance.

*
d
*
= 1/p
^{
17
}
where
*
d
*
= distance of a star in pc and
*
p
*
= parallax angle of that star in arc seconds. Other relationships:
*
1 pc = 3.26 light years (ly)
*
*
= 3.09 × 10
^{
13
}
km,
*

*
= 206,265 AU
*
. (0ne second of arc = one thirty six hundredth of a degree)

Trigonometric parallax is used only for measuring the distance of stars in the solar neighborhood. For more remote stars where
*
p
*
is very tiny other methods for measuring distances are required.

### Stellar Magnitudes

The system used by astronomers to measure the brightness of stars is based on a classification developed by Ptolemy in the second century AD. He compiled a catalogue of several hundred stars and then grouped stars into six categories according to how bright they appeared to the naked eyes. He classified the brightest stars as first magnitude and the dimmest stars as sixth magnitude.

Today astronomers recognize that the first magnitude stars are about one hundred times as bright as the sixth magnitude stars. To establish a more precise scale a star of magnitude 1 is defined as 100 times as bright as a star of magnitude six. A star of n
^{
th
}
magnitude will be exactly
*
x
*
times as bright as a star of magnitude
*
n + 1
*
. To find an approximation for x by using the fact that the first magnitude star is exactly x times as bright as the sixth magnitude star, and by definition a first magnitude star is 100 times as bright as the sixth magnitude star
^{
18
}
. Therefore

*
x
^{
5
}
= 100
*

*
x = 2.51188. . .
*

Each magnitude of brightness corresponds by a factor of approximately 2.5. Therefore a star of magnitude n is about 2.5 times as bright as a star of magnitude
*
n+1
*
.

The apparent brightness
*
(B
_{
a p p
}
)
*
of a star depends not only on the amount of light it emits (intrinsic brightness) but how far away it is from Earth. If the stars could be moved so that they are all the same distance from the Earth the star's

*B*(apparent brightness) would give a good measure of the intrinsic or absolute brightness

_{ a p p }*(B*). This would allow us to compare the brightness of the stars, or to say which star is the brightest.

_{ a b s }
If we know the distance to a star, then using the inverse square law for intensity we can determine how bright the star would be if it were moved to some other distance. If this were done for all stars whose distance is known, we can choose some standard distance (
*
d
_{
s t a n d a r d
}
*
) and calculate how bright each star would be if moved from its actual distance (

*d*) to this standard distance. The inverse square law gives the equations for apparent brightness and absolute brightness as

_{ a c t u a l }
*
B
_{
a b s
}
*
= C / d s t a n d a r d

_{ 2 }and

*B*= C / d a c t u a l

_{ a p p }_{ 2 }where

*C*represent intrinsic properties of the star that do not depend on where the star is located. If equations are divided by equation 1, then it gives

*B*/

_{ a b s }*B*=

_{ a p p }*(*d a c t u a l / d s t a n d a r d)

*2*

At its actual distance a star has magnitude
*
m
*
. This is called the apparent magnitude. The magnitude that a star would have at the standard distance is denoted by
*
M
*
. This called the absolute magnitude. If the star's actual distance is greater than the standard distance (
*
d
_{
a c t u a l
}
d
*
s t a n d a

_{ r }d ), then

*m > M*. This means that the star is brighter at the standard distance. The difference in brightness can be calculated using the magnitude difference factor 2.5, so at the standard distance it would be 2.5 m - M times as bright as a the actual distance. Therefore (d

_{ a c t u a l }/ d s t a n d a r d) 2 =

*2.5*. In this equation it is assumed that the observed brightness of a star depends only on its intrinsic brightness and on how far away the star is located. The effect of the other particles between the observer and the star that would diminish the observed brightness is ignored. Astronomers use the standard distance to be 10 pc ( parsec is magnitude in light years). The equation can therefore be written as (d/10pc) 2 =

^{ m - M }*2.5*m - M , where

*d*is the actual distance of the star.