Maths and Astronomy Why Maths Comenius project

MATHS AND ASTRONOMY

G I M N A Z J U M I M . A N N Y WA Z ÓW N Y , G O L U B - D O B R Z Y Ń

MATHS AND ASTRONOMY

THIS EBOOK WAS PREPARED

AS A PART OF THE COMENIUS PROJECT

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by the students and the teachers from:

GIMNAZJUM IM. ANNY WAZÓWNY

IN GOLUB-DOBRZYŃ IN POLAND

This project has been funded with support from the European Commission.

This publication reflects the views only of the author, and the

Commission cannot be held responsible for any use which may be made of the

information contained therein.

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Astronomy is a fascinating science, from the distances to and inter-workings of stars

and planets. Most of mathematical skills are grade school level arithmetic skills, so it

is not needed to be a mathematics major to understand the mathematical concepts

necessary to do well in an introductory astronomy class. One of the first

mathematical challenges we find ourselves facing in astronomy is dealing with very

large numbers, Basic arithmetic rules of manipulation for addition, subtraction,

multiplication, and division apply to numbers written in scientific notation.

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Determining the distance to objects in astronomy is one of the most important tasks

and yet it is one of the most difficult ones.

In the solar system, even the kilometer is inconveniently small to use as a unit. One

way scientists measure the distance between

planets is to use the distance from the Earth

to the Sun as the standard unit of

measurement. This distance is called an

Astronomical Unit, or 1 A.U. It is equal to

approximately 150,000,000 kilometers.

Incredibly, stars and galaxies are much

farther that even AU's become unwieldy and

light-years become the standard.

A lightyear (ly) is the distance light travels

in one year at its speed of 300,000 km/sec. If

we multiply this speed by the number of

seconds in a year, we get the distance:

A light-year is the distance light travels in

one year – 9.5 trillion km – and that light

travels 300,000 km per second.

Our galaxy, the Milky Way, is about

150,000 light-years across, and the

nearest large galaxy, Andromeda, is 2.3

million light-years away.

Image from: www.mathscareers.org.uk

Image from:

Image from:

Some other distances in light years:

Object Distance in light years

Nearest Star (Proxima Centuri) 4.2

Sirius the dog star (the brightest star in the

sky) 8.6

centre of the galaxy approximately 30 000

If the angular diameter is not very large (which is

typically the case), problems of this type may be

done without the use of trigonometry. For small

enough α, tan α ≈ α when the angle is measured

in a common unit known as radians.

A parsec is defined as the distance at which the parallax angle is 1 arc second. A

parsec is approximately 3.26 light-years.

It is also common to define large multiples of the parsec,

1 kiloparsec (kpc) D 103 pc

1 megaparsec (Mpc) D 106 pc

for very large distances. The following table summarizes some common astronomy

distance units and gives their size in meters.

Quantity Abbreviation Distance (km)

Astronomical unit AU 1.50·108

Light year ly 9.46·1012

Parsec PC 3.08·1013

Kiloparsec kpc 3.08·1016

Megaparsec Mpc 3.08·1019

We should remember that when multiplying numbers in scientific notation, we have

to multiply the number part, times ten to the power of the sum of the exponents.

For example:

When dividing numbers in scientific notation, we have to divide the number part. The

answer is multiplied by 10 to the power which is the difference between the

exponents.

For example:

Examples

1. Convert each number of light years to kilometers.

a) 6 light years

b) light years

c) light years

2. Neptune is 4500 million kilometers from the Sun. How far is this in AUs?

Neptune is 30AU far from the Sun.

3. The second brightest star in the sky (after Sirius) is Canopus. This yeallow-white

supergiant is about 1.141016 kilometers away. How far away is it in light years?

Answer: It’s about light years.

4. Regulus (one of the stars in the constellation Leo the Lion) is about 350 times

brighter than the Sun. It is 85 light years away from the Earth. How far is this in

kilometers?

Answer: Regulus is away from the Earth.

5. Calculate the diameter of the Milky Way galaxy in kilometers assuming that its

radius is 50 ly.

Diameter of the Milky Way galaxy (D):

D = 2 · 50 · 9.46 · 1012 = 9.46 · 1014 km

30 AU

6. The distance from earth to Pluto is about 28.61 AU from the earth. How many

kilometers is it from Pluto to the Earth?

Answer: Pluto is about away from the Earth.

7. A star is 4.3 light years from Earth. How many parsecs is this?

Answer: It’s parsecs.

8. Our Galaxy is 100,000 ly wide. How many meters

wide is it?

Answer: It’s meters.

Image from: www.hplusmagazine.com

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Gravitation is a natural phenomenon by which all physical bodies attract

each other. It is most commonly experienced as the agent that gives weight to objects

with mass and causes them to fall to the ground when dropped.

Gravity is the force with the Earth, Moon, or other massive body attracts an

object towards itself. By definition, this is the weight of the object. All objects on

Earth experience a force of gravity, which is, directed “downward” towards the center

of the object the Earth.

In 1687, English mathematician Sir Isaac Newton

published “Principia”, which hypothesizes the inverse-square

law of universal gravitation.

He wrote: “I deduced that the forces which keep

the planets in their orbs must [be] reciprocally as

the squares of their distances from the centers

about which they revolve: and thereby compared

the force requisite to keep the Moon in her Orb with

the force of gravity at the surface of the Earth; and

found them answer pretty nearly”.

Newton's law of universal gravitation:

Where G is a constant equal to

m = mass of the body 1,

M = mass of body 2,

r = radius between the two bodies.

The force due to gravity is given by

Equating both the force formula we get

Hence acceleration due to gravity formula is given as

It is used to find the acceleration due to gravity anywhere in space. On earth the

acceleration due to gravity is 9.8

.

Examples

a) Calculate the acceleration due to gravity on Earth.

Given:

r =

M =

so

Check the units:

a) Calculate the acceleration due to gravity on the Moon. The Moon’s radius is

and its mass is

Remember,

so

Gravity differs depending on what planet you are on. This is because the planets vary

in size and mass.

b) Calculate the acceleration due to gravity on Mercury. The radius of Mercury is

about 2.43106 m and its mass is 3.181023 kg.

Given:

6

Answer: The acceleration due to gravity on Mercury is

c) Calculate the acceleration due to gravity on Venus. The radius of Venus is

about 6.06 106 m and its mass is 4.88 1024 kg.

Given:

Answer: The acceleration due to gravity on Venus is

Planet Radius (m) Mass (kg) g

Mercury 3.6

Venus 8.9

Earth

Mars

Jupiter 26.0

Saturn 11.2

Uranus 3.61

Neptune 13.3

Examples

1. A 6.2 kg rock dropped near the surface of Mercury reaches a speed of

in 5.0s.

a) What is the acceleration due to gravity near the surface of Mercury?

b) Mars has an average radius of 2.43 m. What is the mass of Mercury?

a)

The acceleration due to gravity is 3.61

b)

The mass of Mercury is

2. In The Little Prince, the Prince visits a small

asteroid called B612. If asteroid B612 has a radius of

only 20.0 m and a mass of 1.00 104 kg, what is the

acceleration due to gravity on asteroid B612?

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Some planets have a stronger gravity than Earth's, some have weaker. On a planet

with a weaker gravity, we would be able to carry more mass and jump higher. When

Astronauts visited the Moon, which has one sixth of our gravity, they bounced around

on the surface as if they were floating with each step. On a planet with a stronger

gravity, we might be forced to our knees by just our own weight.

Weight can be defined by the gravitational force on an object, with a known mass.

This means that in order to obtain the weight of an object we take the mass of an

object, and multiply that by the acceleration of gravity. The mass of an object is the

same no matter where it exists, while the weight of an object changes depending on

what the gravitational field strength where the object exists. Weight is measured in

Newtons (N).

Weight is a force caused by the pull of gravity acting on a mass:

1. The Earth's moon is the only heavenly body

that people have walked on. The gravity of the

moon is 17% of Earth's gravity. To calculate my

weight on the Moon, I multiplied my weight by

0.17.

Answer: My weight on the Moon is about 98N.

2. Mercury is the smallest planet, and the planet closest to the sun. The gravity of

Mercury is 38% of Earth's gravity. To calculate my weight on Mercury, I

multiplied my weight by 0.38.

Answer: My weight on Mercury is about 220 N.

3. Venus is known as the “Cloudy Planet” because it is covered with thick, yellow

clouds. The gravity of Venus is 90% of Earth's gravity. To calculate my weight

on Venus, I multiplied my weight by 0.9.

Answer: My weight on Venus is about 520N.

4. Mars is known as the “Red Planet” because the soil is filled with orange-red

particles. The gravity of Mars is 38% of Earth's gravity. To calculate my weight

on Mars, I multiplied my weight by 0.38.

Answer: My weight on Mars is about 220 N.

5. Jupiter has more moons than any other planet. So far, scientists have

discovered 63! The gravity of Jupiter is 234% of Earth's gravity. To calculate

my weight on Jupiter, I multiplied my weight by 2.34.

Answer: My weight on Jupiter is about 1353 N.

6. Saturn is known as the “Ringed Planet” because it has colourful rings made of

rock and ice. The gravity of Saturn is 108% of Earth's gravity. To calculate my

weight on Saturn, I multiplied my weight by 1.08.

Answer: My weight on Neptune is about 624 N.

7. Neptune is a blue planet with extremely strong winds. The gravity of Neptune

is 112% of Earth's gravity. To calculate my weight on Neptune, I multiplied my

weight by 1.12.

Answer: My weight on Neptune is about 648 N.

Planet/star Mass(kg) Gravity ( relative

to Earth) Weight (N)

Moon 59 17%

Mercury 59 38%

Venus 59 90%

Mars 59 38%

Jupiter 59 234%

Saturn 59 108%

Neptune 59 112%

The gravitational field strength, g, of a planet is the weight per unit mass of an object

on that planet. It has the units N/kg e.g.

Earth g = 9.8 N/kg

Mars g = 3.69 N/kg

Moon g = 1.6 N/kg

The weight of an object can be calculated on different planets,if we know that object's

mass and the gravitational field strength of the planet. We can also calculate weight

using the following formula.

Where F weight and it is measured in Newtons (N)

m mass and it is measured in kilograms (kg)

g gravitational field strength, g, of a planet is the weight per unit mass of an object

on that planet. It has the units N/kg.

Planet/star Mass(kg) Gravitation (m/s2) Weight (N)

Earth 59 9.8 578.2

Moon 59 1.6 94.4

Mercury 59 3.73 220.07

Venus 59 8.87 523.33

Mars 59 3.69 217.71

Jupiter 59 25.9 1528.1

Saturn 59 11.19 660.21

Uranus 59 8.69 512.71

Neptune 59 11.28 665.52

Examples of some calculations:

For Earth:

For Mercury:

For Jupiter:

For Neptune:

1. How much more would you weigh on Jupiter than Earth?

For Jupiter we have:

For Earth we have:

Answer: I would weigh 949.9 N more on Jupiter than Earth.

2. How much less would you weigh on Pluto than Earth?

For Earth we have:

For Pluto we have:

Answer: I would weigh 531 N less on Pluto than Earth.

3. Would you weigh more on the Earth's moon, or on Mercury?

For Earth's moon we have:

For Mercury we have:

Answer: I would weigh more on Mercury.

4. Somewhere you place a 7.5 kg pumpkin on a spring scale.

If the scale reads 78.4 N, what is the acceleration due to

gravity at that location?

Given: Calculate:

Answer: I think it is on the Earth.

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On the average highway, a car travels at

approximately 100 km/hour. How fast

do the planets travel as they revolve

around the Sun?

1. If the Earth day comprises 24 hours we calculate the number of Earth hours

required to complete the orbit of each planet.

2. We know the circumference of each planet’s orbit and the number of hours to

complete that orbit. We use this information to calculate the planet’s speed.

1. For Mercury we have:

t = 88·24 = 2112

The velocity is:

v - velocity

s - circumference of an orbit

t - time to complete orbit

2. For Venus we have:

t = 225 · 24 = 5400

The velocity is:

3. For Earth we have:

t = 365 · 24 = 8760

The velocity is:

4. For Mars we have:

t = 687 · 24 = 16488

The velocity is:

5. For Jupiter we have:

t = 4333 · 24 = 103992

The velocity is:

6. For Saturn we have:

t = 10759 · 24 = 258216

The speed is:

7. For Uranus we have:

t = 30685 · 24 = 736440

The velocity is:

8. For Neptune we have:

t = 60189 · 24 = 1444536

The velocity is:

The following table gives information about the orbits of the eight planets of the solar

system. The force of gravity makes the planets move in orbits that are nearly circular

around the Sun.

Planet Circumference

of orbit (km)

Earth time

to complete

orbit (days)

Earth time

to complete

orbit (hours)

Planet

Speed

(km/h)

Mercury 5.79·107 88 2112 2.7·10⁴

Venus 1.08·108 225 5400 2·10⁴

Earth 1.50·108 365 8760 1.7·10⁴

Mars 2.28·108 687 16488 1.38·10⁴

Jupiter 7.78·108 4333 103992 7.48·103

Saturn 1.43·109 10759 258216 5.5·103

Uranus 2.87·109 30685 736440 3.89·103

Neptune 4.50·109 60189 1444536 3.1·103

Questions:

1. Which planet travels fastest?

Answer: The fastest planet is Mercury.

2. Which planet travels slowest?

Answer: The slowest planet is Neptune.

The closer a planet is to the Sun, the faster it travels in its orbit and the less time it

takes to complete a full trip around the Sun.

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Have you ever wondered how old you would be if you lived on another planet? Using

a simple calculation against the planetary year, we can do just that!

We should remember:

Day: the time taken for the Earth to rotate once about its axis.

Year: the time taken for the Earth to revolve once around the Sun

All the planets in the solar system revolve around the Sun in the same direction. On a

given planet, the “year” is the period of time this planet takes to complete one orbit

around the Sun.

If a year is described as the amount of time it takes for a planet to revolve around the

Sun, for the Earth it’s 365.25 days, then our age would be different on each planet.

A planetary year is the length of time it takes that planet to revolve around the Sun.

The planets revolve around the sun in different amounts of time, so a "year" on each

planet is a different amount of time. The farther a planet is from the sun, the longer

its year.

Using the chart below I can figure out how old I would be.

Planet

Number of

days in a

planetary

year

- revolution

Multiply

your age by Years Days

Mercury 87.97 4.152 61 91

Venus 224.7 1.626 24 22

Earth 365.25 1 14 5401

Mars 687 0.53 7 5251

Jupiter 4.333 0.084 1 13032

Saturn 10.759 0.034 0.5 12312

Uranus 30.685 0.012 - 7499

Neptune 60.188 0.006 - 8025

To work out your age on other planets, first you need to calculate how many Earth

days you have been alive (being careful of leap years). You can then calculate how

many days or years that would be on another planet by finding out how long that

planet takes to spin or orbit.

I was born on 3rd June 1999. Today is: 17 March 2014.

I am 5,401 days old (in days).

Mercury:

Calculations for Saturn:

Calculations for Venus:

For example, if I am 14 years ( 5401 days )

old here on Earth and I want to know my

age on Venus I have to divide the number

of days on the Earth by the number of

days in a planetary year of Venus:

Calculations for Mars:

If we could live on another planet, our birthdays would take place more or less often

depending on the planet’s revolution period (the time taken to complete one full trip

around the Sun). On a few planets, we couldn’t even celebrate our first birthday

because we wouldn’t live long enough to give these planets time to complete one full

trip around the Sun!

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If you throw a rock straight up in the air, eventually it will come straight back down.

If you fire a gun straight up in the air, the bullet will travel higher than the rock but

will also eventually come straight back down.

Escape velocity is defined as a minimum velocity with which a body should be

projected so that it overcomes the gravitational pull of the earth.

What speed is required to escape the pull of Earth’s gravity?

In physics, escape velocity is the speed at which the kinetic energy plus the

gravitational potential energy of an object is zero. It is the speed needed to "break

free" from the gravitational attraction of a massive body, without further propulsion.

If the kinetic energy of an object launched from the Earth were equal in magnitude to

the potential energy, then in the absence of friction resistance it could escape from

the Earth.

This can be written mathematically as:

Re-arranging the equation to find v will give the escape velocity for the Earth.

The escape velocity (vesc) of a body depends on the mass (M) and the radius (r)

of the given body. The formula which relates these quantities is:

Escape velocity formula is helpful in finding escape velocity of any body or planet, if

mass and radius is known. It has wide applications in space calculations.

If a rocket is launched with the velocity less than the escape velocity, it will eventually

return to Earth.

If the rocket achieved a speed higher than the escape velocity, it will leave the Earth,

and will not return.

1. Determine the mass and radius of the planet you are on.

For Earth, assuming that we are at sea level, the radius is 6.38106 meters and the

mass is 5.971024 kilograms.

We need the gravitational constant (G), which is 6.6710-11

. It is required to use

metric units for this equation:

Given:

Radius ( r ) = 6.38106 meters

Mass ( M )= 5.971024 kilograms

Gravitation constant = 6.6710-11

2. Using the above data, calculate the required velocity needed to

exceed the planet's gravitational potential.

Check the units:

3.

The escape velocity of Earth comes to about 11.2 kilometers per second from the

surface.

Example

Calculate the escape velocity of the moon if Mass is 7.35 1022

kg and radius is 1.7 106 m.

Solution:

M = 7.35 1022 kg,

R = 1.7 106 m

Hence Escape Velocity is given by

So we could see that when the Apollo astronauts departed from the surface of the

Moon, they only had to be travelling one fifth the speed they travelled in order to

leave the Earth.

Example

Calculate Mercury and Jupiter escape velocity.

Solution

Given:

Radius ( r ) = 2.4103 km = 2.4106 meters


The escape velocity is given by:

Given:

Radius ( r ) = 7.15104 km = 7.15107 meters


The escape velocity is given by

Answer:

Jupiter's escape velocity is (60/4.3) 14 times higher than Mercury's.

For Mercury

For

Jupiter

A table of masses and radii is given below for many bodies in the Solar System.

Planet Mass (kg) Radius (km)

Mercury 3.301023 2439

Venus 4.871024 6051

Earth 5.981024 6378

Mars 6.421023 3393

Jupiter 1.901027 71492

Saturn 5.691026 60268

Uranus 8.681025 25559

Neptune 1.021026 24764

Example

Calculate the mass of the planet Mars, whose escape velocity is 5 103?

Given:

Radius of Mars (r) = 33.97 105 kg,

Escape velocity (ve) = 5 103,

Mass of the planet mars (M) = ?

Substitute the given values in the formula:

Mars mass:

Answer:

The mass of planet Mars is M = 6.3641023 kg

LINKS:

www.smith-teach.com

www.planetfacts.org

www.chandra.harvard.edu

www.galaxymaine.com

www.astronomy.wonderhowto.com

G I M N A Z J U M I M . A N N Y WAZÓWNY , G O L U B -DOBR Z YŃ

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