Distances in Space Distance Units - the AU - PowerPoint Presentation

Slide1

Distances in SpaceSlide2

Distance Units - the AU

An

Astronomical Unit

(AU) is defined as the average distance between the Earth and Sun.

1 AU =

92 955 807 miles

or

149 597 871 km

AU’s are used to measure distances within the solar system.

There are approximately 63,240 AU’s in one light-year (LY).Slide3

Distance Units - the Light Year

One light year (LY) is the distance light travels in one year.

Light years are not used to measure time.

A light year is a convenient unit to use for measuring distances within the galaxy, especially the distances to nearby stars.

One LY =

5.88 trillion miles

or

9.46 trillion km

.Slide4

Distance Units - Parsecs

A

parsec

is a shortened term for “parallax second of arc”. A star at a distance of one parsec shows a parallax angle of 1 second of 1 degree (1/3600 of a degree). This translates to a distance of 3.26 LY.

Distances in parsecs are not commonly quoted, but diatances in

kiloparsecs (kpc),

megaparsecs (Mpc)

or

gigaparsecs (Gpc)

are often used, especially when referring to distances to faraway galaxies.

1 kpc = 1000 pc = 3260 LY

1Mpc = 1 million pc = 3.26 million LY

1Gpc = 1 billion pc = 3.26 billion LYSlide5

The Distance Ladder

Calculating distances in space is much like climbing a ladder:

Each step higher relies on the previous step below,

and the higher you go, the more uncertain you are about your stability. Slide6

The Distance Ladder

Calculating distances in space is much like climbing a ladder:

Each step higher relies on the previous step below,

and the higher you go, the more uncertain you are

about your stability .

Knowing the distance to one object often serves as a stepping stone to determining the distance to another.Slide7

The Distance Ladder

Calculating distances in space is much like climbing a ladder:

Each step higher relies on the previous step below,

and the higher you go, the more uncertain you are

about your stability .

Knowing the distance to one object often serves as a stepping stone to determining the distance to another.

The most distant measurements are those with the highest degrees of uncertainties.Slide8

The Distance LadderSlide9
Slide10

Preliminary Measurements

The early Greeks had to first determine the diameters of the Earth and Moon before they could calculate a distance to the Moon.

Eratosthenes performed the now famous experiment between Syene and Alexandria to determine the Earth’s circumference, and thus its diameter. For a review of the details of the experiment, check out this link:

http://dev.physicslab.org/Document.aspx?doctype=2&filename=IntroductoryMathematics_EarthCircumference.xmlSlide11

Earth Circumference

A short excerpt of the Eratosthenes experiment from the Cosmos with Carl Sagan explains this idea:

https://www.youtube.com/watch?v=G8cbIWMv0rI

A teachers guide to replicate the activity is here:

http://www.physics2005.org/projects/eratosthenes/TeachersGuide.pdf

In order to really do this, you must find another school or observer that is along your line of longitude, and at a some distance away.Slide12

Size of the Moon

Knowing the size of the Earth, Aristotle used observations from a lunar eclipse to get an estimate of the Moon’s angular size. An interactive of this activity is at this link, but you can easily turn it into a pencil and paper activity:

http://dev.physicslab.org/Document.aspx?doctype=2&filename=IntroductoryMathematics_SizeMoon.xml

Slide13

Distance to the Moon

Once Aristotle knew the Moon’s diameter, he could calculate a distance to the Moon by using the property of similar triangles.

(

You may want to first ask students to experiment with different sized coins to see how far they have to move the coin away from their eye to get it to match the size of the Moon in the sky. (But even with a dime, this can be challenging.)

The following activity suggests using a thumbnail, but using (the width of) a pencil works better:

http://dev.physicslab.org/Document.aspx?doctype=2&filename=IntroductoryMathematics_EarthMoonDistance.xmlSlide14

Distance to the Moon

Once Aristotle knew the Moon’s diameter, he could calculate a distance to the Moon by using the property of similar trianglesSlide15

Distance to the Moon

The Apollo XI. astronauts left a mirror on the lunar surface so that laser beams from Earth could be reflected back. The measurement of the incoming light gives a highly accurate measurement of the Moon’s distance, which varies from 363,104 km – 406,696 km, depending on where it is in its orbit.

The link to the full story is here:

http://science.nasa.gov/science-news/science-at-nasa/2004/21jul_llr/Slide16

Distance to the Moon

16

The lunar laser ranging experiment is still operational and can determine the distance between Earth and Moon to a few centimeters.

The McDonald Observatory in Texas

operates the laser that bounces off

the Moon.Slide17

Distance to the Sun – Aristarchus

http://galileoandeinstein.physics.virginia.edu/lectures/gkastr1.html

When the Moon appears to be exactly half lit, the line from the Moon to the Sun must be exactly perpendicular to the line from the Moon to an observer on Earth. (See the figure below). So, if the quarter moon is observed during the day, one could measure the angle between the Moon and the Sun in the sky (the angle α in the figure). You can then construct a long thin triangle, with its baseline the Earth-Moon line, having an angle of 90 degrees at one end and α at the other, and so find the ratio of the Sun’s

distance to the Moon’s

distance.

This may be solved using either

trigonometry or a scale drawing.Slide18

Distance to the Sun – scale drawing

The Earth-Moon distance is 384 400 km (adjacent side).

Use a scale of 1 cm = 100 000 km for your drawing.

Use a protractor to create angles of 90° and 89.8° on the drawing.

Draw out the opposite and hypotenuse sides until they intersect. From that point, draw a straight line back to the adjacent side to represent the Earth-Sun distance.

Measure this in cm and convert it

to km to get the distance from

Earth to the Sun.Slide19

Distance to the Sun – more to come

There wasn’t any good way to determine a more accurate distance to the Sun until after Kepler had derived his three laws of planetary motion, so we will come back to this idea once Kepler’s Laws have been examined.

Sneak preview:

In 1677, English astronomer Edmond Halley proposed a method for calculating the Earth’s distance from the Sun by using the transit of Venus.

Halley died 19 years before his method could be attempted and proven successful (during the 1761 transit of Venus).Slide20

Kepler’s Third Law

Johannes Kepler derived three laws of planetary motion, which revolutionized how astronomers understood the solar system. His third law, published in 1618, gave a way to measure the relative distance of a planet from the Sun:

P

2

=

A

3

where

P

= planet's orbital period in years

and

A

= the planet’s average distance from the Sun, in AU

Example:

Venus orbits the Sun in .615 Earth years.

What is its distance from the Sun?

.615

2

= A

3

A= .723 AUSlide21

A Transit of Venus

A Venus transit is when Venus crosses exactly between Earth and the Sun. The silhouette of the planet may be viewed moving across the solar disc, like a miniature eclipse.

This alignment is rare, coming in pairs that are eight years apart but separated by over a century.  The most recent transit of Venus was in June of 2012. The next transit of Venus pair occurs in December 2117 and 2125.  Slide22

Distance to the

Sun - transit of

Venus - 1

http://

www.exploratorium.edu

/

venus

/question4.

html

Imagine two different people, at two different locations on Earth, viewing the transit of Venus.

The angle between the two different paths measured from Earth is angle E: Slide23

Distance to the Sun - transit of Venus - 2

From

Kepler's

third law, we know the relative distances of all the planets from the Sun. Case in point: We know that Venus's distance from the Sun is

0.72 times the Earth's distance from the Sun.

This distance relationship also tells us angle

V

, the angle between the two paths as seen from Venus: angle

V

is angle

E

divided by 0.72. (This is true only for small angles, which these are.)Slide24

Distance to the Sun - transit of Venus - 3

In addition to angle V, we also need to know the distance between the two observers on Earth, at points A and B.

Call this distance d

A-B

.

For small angles, tan (1/2 A) = 1/2 tan A, so the distance between Earth and Venus is equal to

d

A-B

/ tan V.

Note that this gives only the distance to Venus, not to the Sun.Slide25

Distance to the Sun - transit of Venus - 4

If we have the distance between Earth and Venus, then the distance from Earth to the Sun may be calculated by a simple ratio:

Venus

.72 AU

=

1.0 AU

Earth

distance

107,000,000 km

x km distance

*X= 148,600,000 km

*This was the distance between Earth and Sun at the time of the 2012 transit.

The

average

distance between Earth and Sun is 149,597,871 km.Slide26

Distance from Radar

In modern times, radar has been used to measure distance to the inner planets and asteroids, as well as comets.

The Arecibo radio telescope

in Puerto Rico is one of the

instruments that is used for

these studies.

The Arecibo telescope has the capability to transmit a high powered-beam of radio waves, as well as to detect their returning echoes.Slide27

Distance from Radar

Although radar may be used to determine distances, its primary function is to create maps of planetary surfaces and to determine the shapes of asteroids and comet nuclei.

Radar image of the Moon’s south polar region.Slide28

Parallax

The parallax technique is used to measure the distances to nearby stars, out to about 500 LY. (The more distant stars have such small motions that an accurate parallax angle cannot be determined for them.)

The baseline of Earth’s orbit is used to measure the

parallax angle.

This is the angular amount of perceived shift of the nearby star against the backdrop of the distant stars.Slide29

Parallax and Parsces

There is a simple relationship between a star's distance and its parallax angle:

d

= 1/

p

The distance

d

is measured in parsecs and the parallax angle

p

is measured in arcseconds.

This relationship is why many astronomers prefer to measure distances in parsecs. Slide30

Star Magnitudes

Many of the subsequent distance techniques rely on knowing star magnitudes.

There are two types: Apparent (m) and Absolute (M)

Apparent magnitude

is how bright the star appears from Earth.

Absolute magnitude

is how bright the star would appear if placed at a distance of 10 parsecs.

So if m < M, the star would be closer than 10 parsecs.

Both magnitude scales work the same way: the more negative the number, the brighter the star.

The Sun has an absolute magnitude of 4.74 and an apparent magnitude of -26.7.Slide31

Spectroscopic Parallax - definition

Spectroscopic parallax is a sort of misnomer, because there is no measurement of a shift in position, as there is in (trigonometric) parallax.

The use of the word ‘parallax’ in this instance refers to using a star’s spectrum to find its distance.

This technique can be applied to any star, assuming that its

apparent magnitude

has been measured and a

spectrum

has been recorded.Slide32

Steps for Spectroscopic Parallax

The pattern of

spectral

lines assigns a

spectral

class to the star, giving it a horizontal position on an H-R diagram.Slide33

Steps for Spectroscopic Parallax

2. The thickness of spectral lines is used to assign a luminosity

class to the star, giving it a vertical position on an H-R diagram.Slide34

Steps for Spectroscopic Parallax

3. From its location on H-R

Diagram, an absolute

magnitude can be

estimated.

This links to an applet that may be used to demonstrate these steps:

http://astro.unl.edu/naap/distance/animations/spectroParallax.htmlSlide35

Steps for Spectroscopic Parallax

The

distance

m

odulus

equation can then be used to calculate

the distance to a star:

d

= 10

(

m - M

+ 5)/5

w

here d is the distance in parsecs,

m is the apparent magnitude of the star and

M is the absolute magnitude of the star

Calculating spectroscopic parallax is one of the most important applications of the H-R Diagram.Slide36

The Distance Modulus

36

In addition to the equation, the distance modulus can be easily applied through the use of a

nomogram

.

Draw a straight line connecting the two magnitudes of a star (M and m).

The distance may be read from where the line crosses the middle axis. Slide37

Standard Candles

Some objects in space have fixed and known absolute magnitudes, and can be used to determine distances by using the distance modulus. Since their intrinsic brightness is known, their apparent magnitude is directly an effect of distance.

Think of it like a 100W light bulb: As the distance between you and the light bulb doubles, the brightness drops off according to the inverse-square law:

Brightness = 1/d

2

Doubling the distance makes the light

bulb appear

¼

of its original brightness.Slide38

Examples of

Standard Candles

Here are some celestial objects that are used as standard candles:

Cepheid and RR

Lyrae

variable stars

Type

Ia

supernovae

Spiral galaxies with measured rotational velocities (Tully-Fisher

relationship)

Gravitationally-lensed quasars

There are other standard candles that are measured using

radio

or

x-

rays wavelengths.Slide39

Cepheid variable stars

Cepheid variable stars may be used to measure the distance to a star cluster or nearby galaxy. This technique is good out to distances of 60 million LY.

They obey a well-defined

period-luminosity

relationship

, discovered in the early 1900’s

by the Harvard astronomer Henrietta Leavitt.

This is now known as the

Leavitt Law.

Other types of variable stars were subsequently discovered with similar patterns, such as RR Lyrae variables. Slide40

Cepheid variable stars

To find the distance to a Cepheid variable star, one must first observe its brightness fluctuations over time and plot a light curve. Then its period and average magnitude may be calculated.

This plot shows a

star with a period of

5 days and an average

apparent magnitude

of 15.6:Slide41

Cepheid variable stars

Cepheid variable stars obey a well-defined

period-luminosity relationship

:

The luminosity can be converted to an absolute magnitude, and the distance

modulus equation is then

used to calculate distance.Slide42

Star Cluster Fitting

Another way to use the H-R Diagram to calculate distance is to “fit” the main sequence of a star cluster to a model or standard star cluster. In this method, it is assumed that all the stars of a cluster are at the same distance away, so when it is fitted, the distance to the cluster is determined.

The model cluster contains stars whose distances are known from parallax and therefore have defined absolute magnitudes.

Slide43

Star Cluster Fitting

In this example, a star cluster is plotted (grey dots) against the model track of a known star cluster (red). The vertical distance is adjusted between the two main sequences so that

they overlap.

The stars in this

cluster show dimmer

apparent magnitudes,

so this cluster must be

farther away than the

model cluster.

Slide44

Star Cluster Fitting

The vertical distance is adjusted between the two main sequences so that they overlap. The difference (Δm) may be used in the distance modulus equation as a substitute for (M-m):

d

= 10

((Δ

m)

+ 5)/5

w

here d is the distance in parsecs

between the two clusters

In the previous example, the

Δm was 5.4 magnitudes, so

d

= 10

((

5.4)

+ 5)/5

= 10

2.08

= 120 parsecs

If the model cluster had a known distance of 80 pc, then the fitted cluster distance is 80 + 120 = 200 pc.Slide45

Star Cluster Fitting

Here’s an interactive activity:

http://astro.unl.edu/naap/distance/animations/clusterFittingExplorer.htmlSlide46

Supernova Type Ia light curves

The light curves from Type Ia supernovae may be used as distance indicators to distant galaxies by using a variation of the distance modulus equation:

d = 10

(m-M+5)/5

Where distance (d) is the distance in parsecs,

m is the peak apparent magnitude of the

supernova and M is the average absolute

magnitude of all “standard candle” Type Ia

supernovae, equal to -19.3.Slide47

Supernova Type Ia light curves

As more data from SN light curves accumulated, it was realized that not all Type Ia SN behaved exactly alike. Below is a refinement technique introduced by Mark Phillips in 1993.

There’s a correlation between the peak brightness of the supernova and the rate at which its brightness declines over a 15 day period after maximum light in the B (blue filter) band. It is known as the

luminosity-decline relation

:

Brighter SN fade more slowly, and fainter ones more rapidly.

The change in blue magnitude in the 15 days following maximum light is

Δ m

15

. Slide48

Galaxy redshifts

Most galaxies show a shift in their spectral lines towards longer (redder) wavelengths. This is primarily due to the expansion of the universe.

By measuring the amount of the shift, the line-of-sight velocity and distance to the galaxy may be calculated

using Hubble’s law.Slide49

Measuring galaxy redshifts

The first step in the procedure is to measure the displacement of certain spectral lines from their original (rest) position.

The

rest wavelength

is the laboratory position of the line. The displaced position will be called the

observed wavelength

.

The redshift (Z) is calculated by this equation:

Z = (observed wavelength/ rest wavelength) - 1

Note: Z should be a positive numberSlide50

Galaxy velocity from redshift

Once Z is known, the velocity (v) in km/sec may be calculated by using this equation below:

c (the speed of light) = 3.0 x 10

5

km/sec

v

=

(Z + 1)

2

-1

c = (Z + 1)

2

+1Slide51

Galaxy distance from velocity

Hubble’s law states that a galaxy’s distance is proportional to its velocity, by this relationship:

Distance (in Mpc) = (cz/H) * ((1 + 0.5 z)/(1 + z))

Where H is the Hubble constant. Present values place this number between 67.3 - 73.8 km/sec/MpcSlide52

Distances to Quasars

The distance to a quasar may be calculated in the same manner as that distance to a galaxy. There is one important difference: Quasar spectra (left) are marked by numerous emission lines, while galaxy spectra (right) contain primarily absorption lines.Slide53

Seeing far away Galaxies: Gravitational LensingSlide54