# Physics Tutorial: Astronomical Measurements and Observations

In this Physics tutorial, you will learn:

• What are the most common methods for measuring long distances in the sky?
• Which method is used to measure short/average/long astronomical distances?
• What are the four units used to measure astronomic distances? What are the conversion factors between them?
• What are the instruments used to obtain information from celestial bodies?
• How big is the visible universe?
• Which optical instruments operate in the spectrum of visible light?
• Which instruments use the invisible light to obtain useful info from cosmos?
• How does a telescope work?
• What is space telescope? What advantages does it have compared to other telescopes?

## Introduction

So far, we have learned that distances in the Universe measured in light years vary from a few seconds or minutes (the distance Earth-Moon = 1.28 s of light, the distance Earth-Sun = 6 min 20 s of light, etc.) to billions of light years (in remote galaxies). However, a quick look at the sky provides little insight into which star is closer to us and which is farther. This is because stars have different sizes - a feature that can deceive our perception. The sky appears as a two-dimensional picture where the dimension of depth is missing. Hence, the measurement of distances in the sky has always been a challenging task for astronomers. Fortunately, in more modern times we have access to a significant number of tools and methods to support measurement, methods that depend on the distance of celestial bodies from Earth and which we will discuss and explain in this tutorial.

Sometimes, it is helpful to make observations for long periods in specific sections of the sky to know how the universe changes over time. This will be another element we are going to discuss in this tutorial.

## Measurement of Distances in the Universe

There are three methods used by scientists to measure astronomical distances in the sky. When ordered according to the distance of application they are: radar method (for short astronomical distances), parallax method (for average astronomical distances) and Cepheid method (for long astronomical distances). Let's take a closer look at each of them.

This method is used to measure distances that are not longer than the dimensions of our Solar System. It works by emitting a light signal towards a celestial body (the distance of which from Earth we want to measure) and then calculating the time needed for the signal to turn back after striking the given celestial body. This method is similar to that of echolocation which we explained when examing the physics of sound waves.

There are specific requirements you must follow when using the radar method. First, the EM signal must be very short, a kind of pulse. This is because the time (duration) being measured is very short and if a longer signal is used it can cause errors in measurement. Second, the frequency of the EM signal must be very high so that it does not interfere with surrounding visible light (the range of frequencies in visible light varies from 400 to 750 terahertz, so the frequency of EM pulse must be much higher).

By using the radar method we can measure the distance Earth-Moon at a precision range within a few centimetres. In addition, we can measure distances from a planet, natural satellite, asteroid etc., from the Earth and between each other. The most important distance we measure using the radar method is the distance Sun-Earth which we use as a measurement unit called the astronomic unit (au). This unit is often used as a reference unit for longer distances so we can understand the distance in relative terms of our position on earth to the sun..

The precision of this method relies on the fact that the times measured are short and the displacement of celestial bodies involved is negligible. Hence, we can assume the light pulse as completing a single cycle, one dimensional round-trip. #### Example 1

An EM pulse emitted from the Earth towards Titan (the largest satellite of Saturn) returns back to Earth in 2.5 hours. What is the actual distance of Titan from Earth in kilometres?

#### Solution 1

We know that the light speed in vacuum is 300 000 km/s. The time needed for the pulse to turn back to the starting position is 2.5 h with corresponds to 2.5 × 60 × 60 = 9000 s. Given that this value is double the time needed for the signal to reach Titan, we obtain, for the distance of Titan from Earth:

dtitan = c ∙ ttotal/2
= 300 000 km/s ∙ 9000 s/2
= 1 350 000 000 km

### b. Parallax method

The parallax method method is based on the change in the observation angle of a star in two different periods of a year due to the revolution of the Earth around the Sun. The stars are considered as unmoveable because they rotate very slowly around the centre of their corresponding galaxy. (In the previous tutorial we explained that the Sun completes one revolution around the centre of the Milky Way in 200 million years.) Hence, the star corresponds to the vertex of the observation angle whilst the light paths in two different periods of the year are the sides of this angle. It is better to make two observations every six months, by doing so, we take the largest angle possible and therefore, we eliminate the maximum errors of margin made during measurements. The best measurement is taken if we make the observations during solstices (December 22 and June 22) at the same time of the day, this is becuase the Earth is at the ends of the large axis of ellipse.

The Earth makes an elliptic trajectory around the Sun, because of this the star appears to make a small elliptic motion in the sky (if we take the Earth as a stationary reference frame). The large half-axis a/2 of this ellipse (passing through the position of the Sun) is witnessed on Earth at the angle p known as parallax (see the figure below). Since stars are very far away from Earth, the angle (parallax) measured is much smaller than the one shown in the figure. No parallax can reach 1" of the angle (the symbol (") stands for second of angle, that is 1/3600 of 10. From geometry, it is known that an angle of 1 degree has 60 minutes and 1 minute has 60 seconds - nothing to do with units of time, just the same kind of division used). The most sophisticated measuring tools today can measure angles up to 0.01".

The equation used to calculate the distance d of a star from Earth is

a/2 = d ∙ tan p

where, in this instance, a/2 acts as the opposite legs of the angle p in the right angled triangle involved and d is the hypotenuse of this triangle. We know from trigonometry that for small angles, we can use approximations.

tan p ≈ p

where the angle p is given in radians. Let's consider an example to clarify this point.

#### Example 2

What is the largest distance we can measure with actual tools using the parallax method?

#### Solution 2

Within the calculations we need to use the smallest possible angle the actual tools can measure. This angle acts as the parallax p of star's observation. We have

p = 0.01" = 1/360000 × 10
= 6.28/129 600 000

The maximum distance from Earth to Sun is 152 098 000 km = 1.52098 × 108 km. This value corresponds to a/2 in the equation of parallax method. Hence, we obtain

a/2 = dmax ∙ tan p
a/2 = dmax ∙ p
dmax = a/2/p
= 1.52098 × 108 km/4.845679 × 10-8
= 3.13884 × 1015 km

When converted into light years (1 light year = 9.4607 × 1012 km), this maximum distance becomes

dmax = 3.13884 × 1015 km/9.4607 × 1012 km/light year
= 3.32 × 102 light years
= 332 light years

Hence, this result is a confirmation of what we said at the beginning, i.e. this method is used to measure average astronomic distances from Earth. The distances of, circa, 120 000 stars have been measured so far using this method. The closest star viewed in the Southern Hemisphere is Proxima Centauri which has a parallax of 0.76813".

Given the small value of parallax, scientists have determined an alternative unit for measuring long astronomical distances called parsec (pc). We have briefly mentioned this unit in the previous tutorials where the conversion factor between parsec and light year (1 pc = 3.26 l.y.) was given. However, we didn't explain the origins of this unit (parsec). In scientific terms, one parsec is the distance that corresponds to a parallax angle of 1 second. This means we now we have four units available for measuring distances in the sky: kilometres (km), astronomic units (au), light years (l.y.) and parsec (pc). The conversion factor between all of them is

1 pc = 2.063 × 105 au = 3.09 × 1013 km = 3.26 l.y.

From the definition of a parsec, we can directly compute the distance of a star in parsecs using the formula

d(pc) = 1/p

#### Example 3

What is the distance of Proxima Centauri from Earth in km?

#### Solution 3

From theory, we know that the parallax of Proxima Centauri is p = 0.76813". Hence, we have

d(pc)=1/p
= 1/0.76813
= 1.302 pc

When converted into km this value becomes:

d = 1.302 pc ∙ 3.09 × 1013 km/pc
= 4.02 × 1013 km

### c. Cepheids method

This method is used for measuring very long astronomical distances. It uses the period-absolute magnitude relationship in Cepheid stars to measure their distance from each other and from the Earth. Thus, the period of apparent magnitude variation is measured from an observer on Earth and the absolute magnitude is calculated. Then, we use one of the forms of equation used to represent the relationship between the three quantities: distance, apparent magnitude and absolute magnitude, that is

log d = 1/5 (m - M) + 1

where d is the distance of the given star from Earth (in parsecs), m is the apparent magnitude and M the absolute magnitude of Cepheid star. The reason why we use Cepheid stars and not other types of stars is because Cepheids are the only types of stars whose period varies periodically with time.

If we are able to find correctly the Cepheid's distance from Earth, this provides useful info about the distance where the galaxy that contains the given Cepheid is. Given our observation ability of Cepehids, we can measure distances up to 10 Mpc (i.e. 10 × 106 pc = 107 pc).

Knowing the apparent and absolute magnitude of stars (if measured through modern devices other than calculating the period of revolution) allows us to use the above equation for other stars such as novae and supernovae as well. Hence, we can calculate distances up to 400 Mpc through the equation of Cepheids Method. Besides it, there are also other method available that allow us to measure distances up to the remotest edges of Universe (3000 Mpc). The visible universe extends up to this distance from the Earth. It includes the celestial bodies and everything else that is detectable through modern tools such as cosmic radiation etc.

#### Example 4

A bright star known as Delta Cephei is a red supergiant. The period of its rotation around itself is 5.4 days, average relative magnitude = 4 and absolute magnitude = -2.9.

1. What is the average distance of this star from Earth?
2. In which galaxy is it?

#### Solution 4

1. The Period is not a relevant numerical information here. It only indicates that the given star is a Cepheid (we can guess this by the star name as well). Consequently we have to use the Cepheid method formula to calculate the distance (in the part b).
From theory we know that the distance of the given star from Earth is calculated by the equation
log d = 1/5 (m - M) + 1
where the distance d is given in parsecs.
In this specific case, we have m = 4 and M = -2.9. Hence, we obtain for the average distance of Delta Cephei from Earth:
log d = 1/5 ∙ [4 - (-2.9)] + 1
log d = 1/5 ∙ 6.9 + 1
log d = 1.38 + 1
log d = 2.38
d = 102.38
d = 239.9 pc
2. At this point, we have to convert the distance from parsecs to light years. Since 1 parsec = 3.26 light years, we obtain the given distance:
d = 239.9 pc ∙ 3.26 l.y./pc
= 779.14 light years

We know that the Milky Way is about 100 000 light years wide (beyond that there is vast empty space before the next galaxy begins), the given star in our exampe is, therefore, inside the Milky Way as the distance from Earth is relatively small, much too small for it to be located in another galaxy.

## Instruments Used for Observation of Sky

Observation of the sky is based on the detection of EM radiation incident from sources that produce and emit it. Through various tool, we are able to detect some of this radiation and then process the data to draw the relevant conclusions including the position, temperature, age, size and many more properties of the stars and other celestial bodies.

People have observed the sky since antiquity yet only the frequencies of visible light have been analysed. It is a fact that EM frequencies include a much larger range of frequencies that those of radio waves to gamma radiation. As a result, despite the long-term attempts, only a small section of the sky (limited to the inside of Milky Way galaxy) has been observed until more modern devices were invented.

EM radiation incident from cosmos is absorbed from the Earths atmosphere. Experiments show that high frequency EM waves cannot penetrate the atmosphere. As a result, the harm caused by such waves is avoided. Without its' atmosphere, life on Earth would be impossible as high frequency EM waves would destroy all living organisms.

Observations of the sky are grouped into two main categories: observations made in the visible spectrum and those made in other spectra of EM radiation. We will explain both of them in the following paragraphs.

### a. Observations made in the visible spectrum. Telescopes

The easiest type of sky observation is made using the naked eye. Obviously, there are limitations in this method including the inability to see beyond visible light frequencies and distances the human sight can extend to. Whilst the first obstacle is overcome only recently, the second one (limited sight) was been addressed centuries ago with the invention of the telescope - a magnifying tool used to see remote objects in the sky normally invisible to the naked eye. This is made possible, not by changing the frequency of EM radiation but, by magnifying the view offered in the sky. Telescopes range from the most basic (used to observe the Sun, Moon, and a few planets of the solar system) to the most advanced ones (built by governments of developed countries) that are used to observe the remotest parts of the visible universe.

The telescope, as a magnifying tool, first appeared in the Netherlands tn October 1608. The national government in The Hague discussed a patent application for a device that aided "seeing faraway things as though nearby." It consisted of a convex and concave lens in a tube. The combination magnified objects three or four times. The government found the device too easy to copy and did not award a patent, but it voted a small award to Jacob Metius and employed Hans Lipperhey to make several binocular versions, for which he was well paid. The telescope and binoculars design has effectively remained the same since their initial conception.

The most convenient telescopes used to observe remote stars are known as "mirror telescopes". The diagram of a mirror telescope is shown in the figure below. The mirror relescope diagram illustrates that light rays are incident from a very large distance (almost from infinity) on a big dimensions concave mirror. As a result, the image produced will be at focus on this mirror. A plane mirror is placed slightly before focus to divert the reflected rays to a converging magnifying lens (eyepiece). In this way, a clear and magnified image is obtained. Newton was the scientist who invented the mirror telescope.

The reason why mirrors are preferred to lenses is that mirrors do not have aberrations (image distortions caused by different thicknesses at the various points of a lens). Hence, we always obtain a sharp image when using mirrors instead of lenses. The lens telescope is made of two main parts: objective (the lens in which the original rays from the object fall first) and eyepiece (the lens on which we see the image). These two lenses are positioned in such a way that the image produced by the object falls slightly before the focus of the eyepiece. From the known formula (see Section 12):

1/do + 1/di = 1/F

where d0 is the distance of the image produced by objective which acts as an object for the eyepiece, di is the distance of final image from the eyepiece and F is the focal length of the eyepiece, we can find the position (and height if we want) of the original object. The operating principle of lens telescope is very similar to that of binoculars, for which we will solve an example now.

#### Example 5

A single eye monocular has a length of 20 cm. The objective has a focal length of Fo = 12 cm and the eyepiece has a focal length of 10 cm. An object is placed at 40 m away from monocular. Calculate the magnification produced.

#### Solution 5

First, we must calculate the position of the image in respect to the objective that later will be considered as the object for the eyepiece. Given that do = 40 m = 4000 cm, we obtain for the first part of solution

1/do + 1/di = 1/Fo
1/4000 + 1/di = 1/12
1/di = 1/12 - 1/4000
1/di = 1000 - 3/12000
1/di = 997/12000
di = 12000/997
di = 12.0361 cm

This image (that acts as an object for the eyepiece) has a distance of

de = 20 cm - 12.0361 cm
= 7.9639 cm

from the eyepiece. Hence, we obtain for the position of final image in respect to eyepiece

1/de + 1/df = 1/Fe

Substituting the known values, we obtain

1/7.9639 + 1/df = 1/10
1/df = 7.9639 - 10/79.639
1/df = -2.0361/79.639
df = -79.639/2.0361
df = 39.11 cm

Magnification M is obtained by dividing df and di. We have

M = df/di
= 39.11 cm/7.9639
= 4.91 times

We must note that this result only represents the magnification in length. The true magnification is 2 dimensional (as we see objects in 2 dimensions when using one eye). Hence, the true (or total) magnification Mt of this monocular is

Mt = M2 = 4.912
= 24.1 times

Telescopes are produced either to observe the sky directly or to generate images. When capturing images, a camera is placed at the eyepiece of the telescope instead of the viewers eye.

Telescopes are usually placed on a fixed support (for example on a tripod) to avoid vibrations and obtain a fixed, sharp and stable image. Moreover, no celestial body remains in a fixed position so it is necessary to change the position of the telescope every a few minutes. For this reason, the moveable base of large telescopes is designed to move in the opposite direction to that of the Earth. This approach esnures that the image remains visible via the eyepiece of the telescope for a long time.

Powerful telescopes are placed inside observatories, these are buildings which typically have half-spherical glass roofs called domes. Telescopes are placed on rotatable tracks and the glass window on the spherical roof is also moveable. This is done to protect the telescope from weather conditions and to obtain a realistic image of the sky. The photo below provides a good example of on of these telescopes observing the Moon. Celestial bodies experience physical phenomena that occur in other EM spectra besides visible light. Consequently, modern tools developed over the past 120 years have been designed to obtain information by means other than direct observation. They act as "windows" that connect us to the world of invisible EM waves including radio waves, IR, UV, X but also gamma radiation.

The first sub-branch of astronomy operating in radio frequencies is known as "radio-astronomy". It was developed between 1932-1936 and allows scientist to study solar photosphere as well as other celestial phenomena. Telescopes that operate at radio frequencies are known as "antenna", in analogy to antennas which are used to receive radio waves in communication.

"Infrared astronomy" was developed during the 1960s. It manifests more technical difficulties than radio-astronomy and is therefore more complicated. The first difficulty arises due to the fact that infrared telescopes must be raised up to the top layers of the atmosphere using balloons or artificial satellites. Second, they must be very cool, close to absolute zero to avoid disturbances caused by the interference of IR radiation originating from the cosmos in addition to that emitted by the telescope itself. IR telescopes allow us to observe interstellar dust and clouds in a given galaxy.

The other types of astronomy based on the study of invisible EM waves use satellites to raise telescopes above the atmosphere to avoid disturbances caused by environmental factors. Today there are UV astronomy, X-rays astronomy and Gamma rays astronomy - all sub-branches developed after the 1960s. Thus, the study of UV spectrum allows us to analyse the spectra of many chemical elements that form various celestial bodies including stars and galaxies. X-ray astronomy detects very hot regions and the environment between two stars that move in pairs, where one of them is a white dwarf, neutron star or black hole. This sub-branch of astronomy is the key factor that allows us to identify black holes in the universe. Gamma astronomy on the other hand, deals with very energetic processes in the universe that are able to emit high frequency radiation. For example, gamma explosions (named so due to their very short interval of occurrence) occur very frequently in the universe are analysed through gamma astronomy to identify the factors causing these explosions and thus, understand better how the universe works.

## Space Telescope

As explained earlier, the technology of artificial satellites allows scientists to use space telescopes to avoid atmospheric turbulences. In this way, they are able to receive images that are impossible to receive directly on the Earths surface. The well-known Hubble telescope, launched into space in 1990, operates in the visible and UV spectrum. The main part of this telescope consists on a spherical mirror of 2.4 m diameter, as shown in the photo below. Another technology used in this regard are the spatial probes. These cosmic machines carry on-board all the necessary tools required for the observation and analysis of celestial objects. They usually make a one-way journey from the Earth to other celestial bodies, they don't return back to the Earth so they continiously transmit the images and videos obtained during their flight. This occurs for several year until they are lost in the space and cannot send information back to the laboratories on Earth.

## Summary

There are three methods used by scientists to measure astronomical distances in the sky. When ordered according to the distance of application they are: radar method (for short astronomical distances), parallax method (for average astronomical distances) and Cepheid method (for long astronomical distances).

The Radar method is used to measure distances that are no longer than the dimensions of our Solar System. It is based on the measurement of the time needed for a short light pulse to make a cycle of one-dimensional round-trip. The Radar method is used to measure distances not longer than Solar System dimensions.

The Parallax method is based on the change in the observation angle of a star in two different periods of year due to the revolution of the Earth around the Sun. The best measurement is taken during solstices (December 22 and June 22) when observed at the same time of the day, as in this case the Earth is at the ends of the large axis of ellipse. The equation used to calculate the distance d of a star from Earth through parallax method is

a/2 = d ∙ tan p

where a/2 acts as the opposite legs of the angle p in the right triangle involved and d is the hypotenuse of this triangle. We know from trigonometry that for small angles, we can use the approximation

tan p ≈ p

where the angle p is given in radians. Hence, we can write

a/2 = d ∙ p

A Parsec (pc) is a unit of distance used to measure very long astronomic distances. In scientific terms, one parsec is the distance that corresponds to a parallax angle of 1 second. There are four units available for measuring distances in the sky: kilometres (km), astronomic units (au), light years (l.y.) and parsec (pc). The conversion factor between all of them is

1 pc = 2.063 × 105 au = 3.09 × 1013 km = 3.26 l.y.

From the definition of a parsec, we can directly compute the distance of a star in parsecs using the formula:

d(pc) = 1/p

The Cepheids method is used for measuring very long astronomical distances. It uses the period-absolute magnitude relationship in Cepheid stars to measure their distance from each other and from Earth. The equation involved in this method is

log d = 1/5 (m - M) + 1

where d is the distance of the given star from Earth (in parsecs), m is the apparent magnitude and M the absolute magnitude of Cepheid star.

Observation of the sky is based on detection of EM radiation incident from sources that produce and emit it. Through various tool, we are able to detect some of this radiation and then process the information to draw relevant conclusions regarding the position, temperature, age, size and additional properties related to stars and other celestial bodies.

Observations of the sky are grouped into two main categories: observations made in the visible spectrum and those made in other spectra of EM radiation. The first category includes observations ranging from those made using the naked eye to modern telescopes operating at visible spectrum, where the most famous are mirror telescopes and lense telescopes.

As for observations made outside of the visible spectrum, they belong to relatively new sub-branches of astronomy developed in the last century. They include:

The first sub-branch of astronomy operating in radio frequencies is known as "radio-astronomy". It was developed between 1932-1936 and allows scientist to study the solar photosphere as well as other celestial phenomena. Telescopes that operate at radio frequencies are known as "antenna" - an analogy to the antennas used to receive radio waves in communication.

"Infrared astronomy" was developed during the 1960s. IR telescopes allow us to observe interstellar dust and clouds in a given galaxy.

Other types of astronomy based on the study of invisible EM waves use satellites to raise telescopes above the atmosphere to avoid disturbances caused by environmental factors. Today there are UV astronomy, X-rays astronomy and Gamma rays astronomy - all sub-branches developed after the 1960s. The study of UV spectrum allows us analyse the spectra of many chemical elements that form various celestial bodies including stars and galaxies. X-ray astronomy detects very hot regions and the environment between two stars that move in pairs, where one of them is a white dwarf, neutron star or black hole. This sub-branch of astronomy is the key factor that allows us to identify black holes in the universe. Gamma astronomy on the other hand, deals with very energetic processes in the universe that are able to emit high frequency radiation.

The Hubble telescope, launched into space in 1990, operates in the visible and UV spectrum. The main part of this telescope consists on a spherical mirror which has a 2.4 m diameter

Another technology used in this way is spatial probes. These cosmic machines carry all the necessary tools on-board for the observation and analyse of celestial objects. They usually make a one-way journey from Earth to other celestial bodies and transmit images and videos obtained during their flight. This occurs for several year until they are lost in space and can no longer send information to laboratories on Earth.

## Astronomical Measurements and Observations Revision Questions

1. The parallax of Alpha Centauri - one of the closest stars to Earth is 0.75". How many light years is it from Earth?

1. 4.35
2. 3.26
3. 1.33
4. 0.75

2. The absolute magnitude vs period of rotation graph of Cepheid stars is shown in the graph below. A Cepheid has a period of change in illumination equal to 10 days when viewed from Earth and an average apparent magnitude equal to 5. What is the distance (in parsecs) of this Cepheid from Earth (to the closest integer)?

1. 28
2. 280
3. 631
4. 2057

3. How big is the visible Universe in light years? Assume the visible universe as a sphere with Earth located at centre of this sphere.

1. 6 000 000
2. 3 000 000 000
3. 6 000 000 000
4. 19 560 000 000