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Physics Lesson 22.7.1 - Measurement of Distances in the Universe
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Welcome to our Physics lesson on Measurement of Distances in the Universe, this is the first lesson of our suite of physics lessons covering the topic of Astronomical Measurements and Observations, you can find links to the other lessons within this tutorial and access additional physics learning resources below this lesson.
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.
a. Radar method
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:
= 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
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.
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
= 1/360000 × 2π/360 rad
= 6.28/129 600 000
= 4.845679 × 10-8 rad
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 ∙ 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
= 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
From the definition of a parsec, we can directly compute the distance of a star in parsecs using the formula
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
= 1/0.76813
= 1.302 pc
When converted into km this value becomes:
= 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
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.
- What is the average distance of this star from Earth?
- In which galaxy is it?
Solution 4
- 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 equationlog d = 1/5 (m - M) + 1where 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 - 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.
You have reached the end of Physics lesson 22.7.1 Measurement of Distances in the Universe. There are 3 lessons in this physics tutorial covering Astronomical Measurements and Observations, you can access all the lessons from this tutorial below.
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