Astronomy

Syncing time for measuring the parallax of Mars

Syncing time for measuring the parallax of Mars

I read somewhere (I can't remember the website) that when Cassini and Jean Richer measured the parallax of Mars, they used Jupiter's (or maybe Saturn's) moons to sync their time. How was this done?


Parallax method

Parallax is a method of measuring distance to an object. Similar to how our binocular vision helps us determine distance, the direction to a distant point is slightly different from two separate observation positions. If the distance between the observation positions is known, and the angle between them can be measured, it is a matter of simple geometry to calculate the distance to the object. It is a relatively simple concept but is one of the most important for making astronomical observations.


The Mission of the John J. McCarthy Observatory is to further science literacy, both for students and adults of the Western Connecticut region. With the great appeal of Astronomy, it is a natural tool for creating scientific curiosity and reinforcing interest and skills in science.

The Earth’s opposition with Mars on August 28, 2003 occurred only 41 hours before the Mars perihelion, reportedly the closest we have been to Mars at opposition in some 60,000 years.

This was the optimum time to measure the parallax angle of Mars from Earth, in order to calculate the distance to the Sun as first done by John Flamsteed in 1672.

This article describes the most recent efforts reproduce the experiment performed by John Flamsteed almost 450 years ago. The effort demonstrates the remarkable technique and skill that was required of Flamsteed to make such measurements with the technology available at the time.

The work reported in this article is a continuation of the project initiated by Dr. Moreland in 2001, using an improved micrometer eyepiece, and taking measurements with Mars at its optimum position relative to the Earth. Based on the parallax measurements of Mars, taken on September 16-17, 2003, the astronomical unit (the distance from the Earth to the Sun) was calculated to be 94.2 million miles +/- 9 million miles.


Syncing time for measuring the parallax of Mars - Astronomy

How Do We Tell Distance By Parallax?

Materials:
1. Images

Parallax Diagram LHS illustration LHS
Measuring Parallax LHS illustration LHS
Zenith LHS illustration LHS

LHS: Alan Gould, Lawrence Hall of Science, University of California, Berkeley

2. (Optional) Parallax Drawing Sheet
One for each student.


Presenter's Script:

Measuring distances by parallax depends on noting how the position an object seems to change when you change your point of view.

As the Earth travels around the Sun each year, we change our point of view by over 300 million kilometers! That’s the kind of baseline we need to see the parallax of a star. When astronomers take pictures of the same region of sky six months apart, some stars—the closest ones to us—change position!
To model this, hold your model star at arm’s length. Imagine that your nose is the Sun and your eyes represent the Earth at two points in its orbit, six months apart. To “take a picture” of the sky six months apart, look first through one eye and then the other.
The parallax of your star is amount that it seems to jump against the background stars. The greater the jump, the closer the star is to Earth.
Astronomers tried to use parallax on the stars in the 15th, 16th, 17th, and 18th century: and they all failed. Even with the enormous baseline of the diameter of the Earth’s orbit, the parallax to the brightest (and presumably nearest) stars always came out to be 0 degrees. They were back to having to assume all the stars were infinitely far away.
Finally in 1838 Friedrich Bessel, after a year and a half of observations of one star, 61 Cygni, succeeded in measuring the parallax to a star. The parallax was less than one 10-thousandth of one degree! No wonder it was so hard to measure. Doing the same kind of calculation and graphs that you used, Bessel calculated that 61 Cygni, one of the nearest stars to us, was an incredible 25 TRILLION kilometers away. That’s 25 thousand billion kilometers!
Astronomers have found that the star which makes the biggest jump every six months is Alpha Centauri. It is “only” 10 trillion kilometers away.

We can measure parallax angles by using these degree markings. There would be 360 degree marks in the complete circle. If your star seemed to jump all the way across the sky, from horizon to horizon, its parallax angle would be 180°.
Hold your star at arm’s length, and with one eye closed, line it up so that the top of it is at the center of the bull’s eye when you look at it through one eye. Without moving your star or your head, note how many degrees the top of the star shifts when you look first through one eye and then the other.
That is the parallax angle of your star.
Do not use the left side or right side of the star for this measurement. The easiest method is to use the “centerline” of the star for measuring the shift. For centerline, you can use either the small stick that comes out the bottom, or the “top” point of the star. Please remember your parallax angle. To calculate how far away your star is, you need a worksheet.
Hand out a “How Far Is The Star” worksheet to each student. Increase light so students can read. If students have pencils, they can write their parallax angles down somewhere on the worksheet.
Refer again to Image 12, Measuring Parallax of a Star. Point to parts of the triangle as you explain.
In measuring distance by parallax, the shift in the observing position is the baseline. You used one eye and then the other to shift your observing position, so your baseline was the distance between your eyes—only a few centimeters. Remember in your model, the distance between your eyes represents a baseline that is the diameter of the Earth’s orbit. The sight lines from your eyes to the star form the sides of the triangle. Knowing the length of the baseline and the parallax angle, we know the keys to the unique triangle that is formed by the two observing positions and the star. The height of that triangle is the distance to the star.
Before you can do your calculation, you need to know the baseline that is distance between your eyes. There are three drawings on the worksheet for three different baseline eye separations of 5 cm, 6 cm, and 7 cm.
Please work with a partner to measure the distance between the pupils of your eyes.
Hand out tape measures or rulers.
Now that you know the baseline and the parallax angle, you can find the distance to your star. Use the diagram that is for your eye separation, the left one for eyes 5 cm apart, the middle one for eyes 6 cm apart, and the right one for eyes 7 cm apart. Then simply see what distance corresponds to the parallax angle that you measured.
Your estimate should be pretty close.
Check it with your partner by actually measuring the distance from your eye to your star with the tape measure (or ruler). How close were you?
With real stars, there is no ruler long enough to make a check like this. We have to believe our measurements and calculations.
Collect model stars and rulers/tape measures.
[End of optional section. Note: This entire activity can be done outside the planetarium, after the program. It works well as part of the “How Far Is It?” classroom activity on pp. 38‑42.]


Syncing time for measuring the parallax of Mars - Astronomy

Student& data should be recorded and presented in group format for two reasons so that error analysis can be discussed and so that students notice that measurements can be made from any location. Students should also note that the further the observer from the meter stick the smaller the measured angle. In fact, some students may not notice any change in angle at all. This fact is also important as it shows the limits to parallax measurement. These students should be encouraged to find a way to measure the meter stick's parallax at that distance, i.e. allow the observer to make left and right eye observations a few feet right and left of their baseline. Students should not be allowed to directly measure the distance between their point of observation and the meter stick, since this is not possible for astronomers measuring the distance to stars. Instead, students should be encouraged to reason out the accuracy of their results, perhaps by comparing their results to those of students who observed farther away from and closer to the meter stick and making sure the results are consistent.

One might wonder why parallax measurement is useful since it is certainly easier to measure the distance to the meter stick directly than with parallax. Unfortunately, since the nearest star is 4.3 light-years away, astronomers cannot actually measure the distance to the star in the same way we can measure small distances and for this reason parallax becomes useful.

Parallax can be used to measure the distance to nearby stars since they can be measured against background stars, which do not appear to move as far, or in fact at all, due to parallax since they are so far away. This is because, as the students will have discovered during the activity, the farther from the "star" they are, the smaller the measured angle is on the background. After a star is far enough away, the angle is so small that even if telescopes were on opposite sides of the Earth they would not be able to measure the angle. In this case, the astronomers use the orbit of the Earth to increase the distance between right and left measurements. This is illustrated in the following diagram:

This diagram shows three observations that are taken at three different times during Earth's orbit to measure the parallax of a nearby star against three background objects: a spiral galaxy 12 , an elliptical galaxy 13 and a barred spiral galaxy 14 .


Syncing time for measuring the parallax of Mars - Astronomy

As shown in the following diagram, the observer measures the altitude in relation to the visible horizon from his position at O on the Earth’s surface. So, the observed altitude is the angle HOX. However, the true altitude is measured from the Earth’s centre in relation to the celestial horizon and is the angle RCX.

Point O will be approximately 6367 Km. from the centre of the Earth and so it would seem that the visible horizon is bound to be slightly offset from the celestial horizon. Because of the vast distances of the stars and the planets from the Earth, we can assume that, in their cases, the celestial horizon and the visible horizon correspond with very little error. However, in the cases of the Sun and the Moon, which are relatively near, a correction called Parallax must be added.

Parallax. We measure the altitude of a celestial body from our position in relation to our visible horizon this is known as the observed altitude. However, when calculating the true altitude, measurements are made from the Earth’s centre in relation to the celestial horizon. The displacement between the observed position of an object and the true position is known as parallax.

Parallax corrections for stars and planets. Because the stars and the planets are at such great distances from the Earth, we can assume that, in their cases, the celestial horizon and the visible horizon correspond with very little error. However, in certain cases when extreme accuracy is needed, parallax corrections for Mars and Venus are required and these are listed in the altitude correction tables.

Parallax corrections for the Sun and the Moon. Because the Sun and the Moon are relatively close to the Earth, parallax will be significant and so a correction has to be made. These corrections are included in the altitude correction tables and therefore do not have to be applied separately.

Horizontal Parallax. Parallax error is greatest when the celestial body is close to the horizon and decreases to zero as the altitude approaches 90 o . It is negligible except in the case of the Moon which is close to the Earth in comparison with the other celestial bodies. Because horizontal parallax is significant in the case of the Moon, a separate correction (abbreviated to HP) has to be applied.


Tycho Brahe's Observatory

He built a very elaborate observatory at Ven that made elaborate use of siting tubes but no telescopes since they hadn't been invented yet!

He essentially used a series of long "sextans" that could measure fairly accurately the angle of a star above the horizon. The azimuth (the direction parallel to the horizon) was measured in degrees from some starting point in this circular observatory out portals that were equally spaced around the floor.


His observatory allowed him to take relatively precise data for measuring the position of Mars night after night.


However the resolution (positional accuracy) of his observatory was far too poor to measure tiny parallax shifts in nearby stars.

Upon failing to detect stellar parallax he proposed this strange hybrid model for the solar system which actually opens up more questions. Fortunately, this model didn't last very long.


Tycho Brahe's universe: The Earth is at the center Planets orbit the Sun which in turn orbits the Earth.

Overall contributions of Tycho:

    He made the most precise observations that had yet been made by devising the best instruments available before the invention of the telescope.

This was a "star" that appeared suddenly where none had been seen before, and was visible for about 18 months before fading from view. Since this clearly represented a change in the sky, prevailing opinion held that the supernova was not really a star but some local phenomenon in the atmosphere (remember: the heavens were supposed to be unchanging in the Aristotelian view). Brahe's meticulous observations showed that the supernova did not change positions with respect to the other stars (no parallax). Therefore, it was a real star, not a local object. This was early evidence against the immutable nature of the heavens.

    the earth was motionless at the center of the Universe, or

Not for the only time in human thought, a great thinker formulated a pivotal question correctly, but then made the wrong choice of possible answers: Brahe did not believe that the stars could possibly be so far away and so concluded that the Earth was the center of the Universe and that Copernicus was wrong.


Activity

Determining the position of a star or other object in space is an important concept in astronomy. During this activity you will learn how the distances to nearby stars can be measured using the parallax effect, and put this method into practise to determine the distance to nearby stars.

Planning

Materials List

Preparation

Before the lesson, you will need to locate a suitable area to create your parallax diagram such as a playground that will not be disturbed. Ensure you have all the materials necessary and follow the steps below:

  1. Draw a straight line 24-metres long with chalk. Mark the centre of the line with a circle. This will represent the Sun.
  2. Measuring along the line to one side of the Sun, draw circles at 1-metre, 6-metres and 10-metres. These circles will represent Planet 1, Planet 3 and Planet 5 consecutively.
  3. To the other side of the Sun, draw circles at 5-metres, 8-metres and 12-metres. Label these as Planet 2, Planet 4 and Planet 6, consecutively.
  4. Draw a new 30-metre line extending from the Sun at a perpendicular 90-degree angle. Mark circles at the following points: 9-metres, 15-metres, 21-metres, 26-metres and 30-metres. These points will represent the distant stars, label them Star A-E.

Background

When looking at the night sky, it’s fairly easy to measure how the stars differ in brightness, but discerning how distant the stars are is a more difficult task. One way to measure astronomical distances is called Parallax.

Parallax is the apparent change in position of a nearby object caused by a change in the observer’s point of view. This is demonstrated in the diagram below as the observer (the car) moves forwards and backwards between two positions, it would see the same tree but it would appear to move against the distant background.

To calculate distances in space, astronomers measure the parallax angle as demonstrated on the diagram below:

Once the parallax angle has been measured twice, from opposite sides of Earth’s orbit around the Sun, we can calculate the distance (d) to stars with the parallax method we use the following equation:

d = 1/p

The parallax angle (p) is measure in arcminutes (arcmin) and arcseconds (arcsec) . Just as an hour is divided into 60 minutes and a minute into 60 seconds, a degree is divided into 60 arcminutes and an arcminute is divided into 60 arcseconds.

1 degree = 1° = 1/360 of a circle

1 arcminute = 1' = 1/60 of a degree

1 arcsecond = 1" = 1/60 of an arcminute = 1/3600 of a degree

The parallax method can only be used to measure the distance to stars close enough to show a measurable parallax, this stretches to about 100 parsecs (or

Instructions

In this activity you will investigate how the distance of an object is related to how far it appears to move when you view it from different perspectives and discuss how parallax can be used in astronomy to reveal the distance to nearby stars.

1. Ask the students if they know what the term “parallax” means. Explain that it is the apparent change in the position of a nearby object that is actually caused by the movement of the observer. This may seem like a foreign concept to some students, so explain that this is something that you use constantly, even though you might not notice it.

2. Place a sticker somewhere in the room. Ask the students to extend their arm, close one eye and cover the object in their vision with their thumb. Then instruct the students to keep their hand still and switch eyes (close the other eye). Ask them to switch back and forth several times. Discuss the following questions:

  • What did you notice?
  • Did your thumb physically move?
  • How did your thumb appear to move?
  • What could explain your thumb’s apparent movement?

3. Ask the students to move their thumb close to their face while switching eyes. Ask them the following questions:

  • How does the apparent motion of your thumb change?
  • What is it about your eyes that allows you to see this?
  • If you could make your arms longer, would you expect the apparent shift to increase or decrease? Why?

4. Explain to your students that the change in their thumb’s apparent position relative to the background object is due to a change in the viewer’s position. In this case, the “viewer” is your left or right eye. The few centimetres of separation between your eyes means that their viewing positions are different.

5. Now explain to your students that parallax allows us to find the position of distant objects such as stars, since just like our eyes, Earth changes position every 6 months as it orbits the Sun.

Parallax can be used to measure the distances to nearby stars. As the Earth orbits the sun, a nearby star will appear to move against the more distant background stars, in the same way the tree appears to move against the more distant mountains in the diagram above.

Astronomers can measure a star's position twice in a year, with a 6-month gap between the first and second measurement, and calculate the apparent change in position. The star's apparent motion is called stellar parallax.

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

d = 1/p

The distance d is measured in parsecs, the parallax angle p is measured in arcseconds. The radius of Earth’s orbit is 1 AU (149, 598, 000 km).

1. Ask students to gather into groups (no more than 5 students per group) and lead them outside to the Parallax Diagram.

2. Assign each team to one of the planets and provide them with an Astrolabe, Instruction Sheet and a Worksheet.

3. Ask the groups to use the Parallax Diagram to work through part one of the Student Worksheet

4. When they have finished part one of their worksheets, gather the groups and head back into the classroom.

1. Open the Parallax table spreadsheet on your computer and use a projector to allow the class to see it for the class to see.

2. Ask each group, one at a time, to provide the parallax angle for each star as seen from their planet. Leave this data on the screen for the class to see and ask them to complete part two of their activity worksheet.

3. Once the students have worked through the second part of their worksheet move on to the discussion section.

1. Ask the students what they notice about the results – did each team end up the same distance? Is there a correlation between percent error and baseline distance (distance from the Sun)?

2. If each group measured the parallax angle with the same amount of care, there should be a notable trend showing teams assigned to planets further from the Sun getting more accurate results. Ask students why they think this is.

The answer is that as the angle gets closer to 90-degrees, inaccuracies are more exaggerated by the tangent function. This means that a very long baseline is advantageous for parallax measurements.

3. Ask students what is the longest possible baseline for Earth-based parallax measurements? The answer is 2 AU, or twice the distance between the Earth and the Sun.

4. Do the students notice a correlation between the distance of a star and the percent error?

For the reason stated previously, the further a star is the less accurate its distance calculations will be on average. This is one of the problems with the parallax method for measuring distances in the cosmos.

5. Ask the students to fill in part three of their worksheet before discussing the problems with parallax in further detail.

Evaluation

It sounds fairly easy to measure the distance to another star just make two measurements of its position six months apart.

In practice, however, it is very difficult. The first successful measurement of stellar parallax came more than two hundred years after the invention of the telescope.

Ask your students to discuss some possible difficulties of measuring cosmic distances using parallax. Ensure they mention the following points:

Parallax shifts are always small.

Parallax shift is even smaller than the apparent size of the star. In additional, starlight is refracted by Earth's atmosphere and causes the star to appear blurred. Determining the position of a star, plus that of several reference stars in the same field, to a very small fraction of this blurry dot is not an easy task.

All stars in a field exhibit parallax.

In practice, astronomers usually measure the shift of one star in an image relative to other stars in the same image. However, as the Earth moves from one side of the Sun to the other, we will see all the stars in the field shift, not only those of interest.
Astronomers must pick out a set of reference stars that happen to be much farther away than the target star. Distant stars will shift by a much smaller angle, so by measuring the position of the nearby target star relative to those distant ones, it is possible to detect its minute shift.

Limitations of Distance Measurement Using Stellar Parallax.

Parallax angles of less than 0.01 arcsec are very difficult to measure accurately from Earth because of the effects of the atmosphere. This limits Earth-based telescopes to measuring the distances to stars about 1/0.01 or 100 parsecs away. Space-based telescopes can get accuracy to 0.001, which has increased the number of stars whose distance could be measured with this method. However, most stars even in our own galaxy are much further away than 1000 parsecs, since the Milky Way is about 30,000 parsecs across. The next section describes how astronomers measure distances to more distant objects.

Next Steps

Since the parallax method cannot be used beyond 100 parsecs, have your students look up other possible techniques for measuring more distant astronomical objects. Examples include the use of Cepheid Variable stars and Type 1a supernovae for nearby galaxies and redshift for those further away.


Syncing time for measuring the parallax of Mars - Astronomy

One of the few reliable methods astronomers may use to determine the distances to objects in space is called parallax. It is based on trigonometry, so let's start with that.

There are special mathematical tools one can use to determine the lengths of the sides of a right triangle. In the figure below, suppose that we want to know the distance L.

If we can measure the side B, and the angle alpha, then we can use the equation to calculate L.

We can use the same method for a situation in which there are two such triangles, back-to-back:

We can still figure out the distance L with trigonometry. Once again, we calculate

but we could also describe this as

Using trigonometry to measure the distance in a hallway

Your job is to devise some measurement that will permit you to calculate the distance to the tripod. Keep in mind the diagrams above.

Parallax and distance measurements

Measuring angles is difficult (as you may have discovered from the tripod exercise). It's not an easy thing to determine exactly in which direction you are looking at a particular moment. Remember that astronomers, sitting on the Earth's surface, are constantly being rotated around the Earth's axis (once every 24 hours), and carried along with the Earth around the Sun (once every 365 days). It's very hard for us to keep track of the absolute direction of our telescopes when we look at stars or planets.

There are times when it helps to use parallax to determine relative angles. Relative measurements, in astronomy and in all of physics, are often much easier to make than absolute ones. Parallax is simply the apparent shift of a nearby objects relative to objects behind it as one moves from one end of a baseline to another: to see it, hold your arm outstretched in front of you, and hold one finger straight up. Hold your arm steady. Close one eye, and look past your finger at some distant object. Now close the OTHER eye and look past your finger again. You should see that it appears to move, relative to the distant object.

Astronomers can do the same thing. If they can observe the same nearby object from two different places, they may detect a difference in its appearance relative to background stars.

Suppose that two astronomers separated by a distance D observe an asteroid simultaneously. One sees the asteroid next to star X, and the other sees it next to star Y. If the astronomers know the angle between those two stars, and they know the distance between their observatories, they can calculate the distance L to the asteroid:

Using parallax to determine the distance to Mars

Make your observations simultaneously at 6:42 AM, Eastern Daylight Time. Note that Sky Map Pro will try to change the time if you move to a different time zone.

Zoom in close to Mars, so that the field of view is about 10 or so arcminutes. Set the limiting magnitude of the chart so that it shows stars down to fifteenth magnitude. You should see a faint star very close to the planet. What star is that?

Use the program to measure the separation between Mars and the star, as seen from each location. Write down your measurements. They will be in polar form: each consists of a distance (in arcseconds), and an angle.

Convert these measurements into rectangular form: the distance in arcseconds to Mars' east (x-direction), and the distance in arcseconds to Mars' north (y-direction). Write these values down, too.

Use the difference in x- and y-directions to calculate the total angle, in arcseconds, by which the star moved relative to Mars. Write down this angle in arcseconds, and convert to degrees.

Make a picture on graph paper which shows the relative positions of Mars and the star, as seen from the different locations. Indicate the size of Mars' disk on your picture. What's really happening is that, seen from different locations on Earth, Mars' position is moving slightly relative to the star. But it is equivalent, and easier to draw, if you pretend that Mars is remaining in place and the star is moving.

Calculate the distance of the baseline between the two cities. Use the baseline, and the angle, to calculate the distance from Mars to the Earth on April 24, 1999.

How does your value compare to the true distance between Mars and the Earth on that date?

Last modified May 9, 1999, by MWR.

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.


All About Astronomy

Exercise questions

I posted here some exercise problems that you can use to expand your knowledge in some chapter in basic astronomy :

  • (a) Of all the natural satellites in the Solar System only the Moon always turns the same face towards its primary.
  • (b) The mass of a planet in the Solar System can be determined only if it possesses one or more satellites.
  • (c) The planet with the largest apparent angular diameter when nearest the Earth is Venus.
  • (d) Pluto is the planet farthest from the Sun.
  • (e) A lunar eclipse may occur if the Moon is new.

2.) Calculate the mean density of Jupiter from the following data, assuming the orbits of Earth and Jupiter to be circular and coplanar:

  • Angular semi-diameter of Jupiter at opposition = 21”,8
  • Orbital radius of Jupiter = 5,2 A.U.
  • Mass of Jupiter/mass of Earth = 318
  • Mean density of Earth = 5,5 kg/m^-3
  • Sun’s horizontal parallax = 8”,8

3.) The two components of a binary star are approximately equal brightness. Their maximum separation is 1”,3 and the period is 50,2 years. The composite spectrum shows double lines with a maximum separation of 0,18 Angstrom at 5000 Angstrom. Assuming that the plane of the orbit contains the line of sight, calculate (i) the total mass of the system in the terms of the solar mass, (ii) the parallax of the system.

Basic Astronomy (part 1)

Before we learn further about astronomy, there are some basic knowledges that we must know and understand.

First, we will talk about measuring distance in astronomy.

Astronomical object lies in a very great distance from us. So far than our sense can perceive. That’s why our sense can’t have a 3-D visualization of the universe. Our sense can’t differ closer to farther objects. So, we need some trick to know how far an object from us. One of the simplest method used by astronomers to measure distance of some closest star is using the parallax effect.

Parallax is an optical effect seen when the observer seeing an object from two different positions. The object will be seen shifted relative to the farther background objects.

The parallax effect is one of those things you see everyday and think nothing of until it’s given some mysterious scientific-sounding name. There’s really no magic here. Consider the following simple situation.

You’re riding in a car on a highway out west. It’s a beautiful sunny day, and you can see for miles in every direction. Off to your left, in the distance, you see a snow-capped mountain. In front of that mountain, and much closer to the car, you see a lone ponderosa pine standing in a field next to the highway. I’ve diagramed this idyllic scene in the figure below:

As you drive by the field, you notice an interesting sight. When you’re in the position on the left side of the figure, the tree appears to be to the right of the mountain. You can see this in the figure by the fact that the line of sight to the tree (indicated by the green line) is rightward of the line of sight to the mountain (indicated by the blue line). A picture of what you see out the window of your car is shown below the car.

The interesting part is that as your drive on, you notice that the tree and mountain have switched positions that is, by the time you reach the right hand position in the above figure, the tree appears to be to the left of the mountain. You can see this in the figure by noting that the line of sight to the tree (green line) is leftward of the line of sight to the mountain (blue line). A picture of what you see out the window of your car now is shown below the car.

What’s going on here? It’s pretty clear that the tree and mountain haven’t moved at all, yet the tree appears to have jumped from one side of the mountain to the other. By now, you’re probably saying “Well, DUH, the tree is just closer to me than the mountain. What’s so remarkable about that?” I would answer, “There’s nothing at all remarkable about it. It’s just the effect of parallax.” In fact, if you understand the above discussion, you already understand the parallax effect.

Now let’s talk about measuring the distance to the tree using this information. From the above information, you can see that it would be pretty easy to measure the angle between the direction to the tree and the direction to the mountain in both instances. Let’s call those angles A and B, respectively. Now, if the mountain is sufficiently distant so that the direction to the mountain from both viewpoints is the same, then the two blue lines in the figure below are parallel.

This helps a lot, because we can then show that the angle made by the two green lines (i.e., the difference in the direction to the pine tree from the two viewpoints) is equal to the sum of A and B. To see this, construct a line through the pine tree parallel to the two blue lines in the figure (this line is shown as a dotted line above). Then all of the blue lines are parallel, and each of the green lines crosses a pair of parallel lines. Reach deep back into your high school geometry (or equivalently, just stare at the above figure for a minute), and you’ll remember or realize that the angles at the pine tree labeled A and B have the same values as the angles A and B measured at the two car positions. Thus, the angle between the two green lines is the sum of A and B, which are angles we can measure from the comfort of our car.

Now, if we know the distance D we’ve traveled, then we have an Observer’s Triangle and we can solve for the distance to the tree using the Observer’s Triangle relation

alpha/57.3 = D/R where alpha is the angle at the tree (A + B), D is the distance we’ve traveled between views, and R is the distance from the road to the tree. (source : Astronomy 101 Specials: Measuring Distance via the Parallax Effect).

We will use the same method to measure the star’s distance. This method is called trigonometric parallax because we only use simple triangulation to find the distance. The only problem is star’s distance is so huge so the parallax effect will be so small (less than 1 arc second 1 arc second = 1/3600 of a degree). So, that’s why this method can only measure accurately for several nearby stars. Farther star will need different, more complex, indirect method to derive its distance.

As explained before, the stars are so far away that observing a star from opposite sides of the Earth would produce a parallax angle much, much too small to detect (That’s why ancient people can’t detect this shifting to prove heliocentric view) . As a consequence, we must use large a baseline as possible. The largest one that can be easily used is the orbit of the Earth. In this case the baseline is the mean distance between the Earth and the Sun—an astronomical unit (AU) or 149.6 million kilometers! A picture of a nearby star is taken against the background of stars from opposite sides of the Earth’s orbit (six months apart). The parallax angle p is one-half of the total angular shift.

However, even with this large baseline, the distances to the stars in units of astronomical units are huge, so a more convenient unit of distance called a parsec is used (abbreviated with “pc”). A parsec is the distance of a star that has a par allax of one arc sec ond using a baseline of 1 astronomical unit. Therefore, one parsec = 206,265 astronomical units. The nearest star is about 1.3 parsecs from the solar system. In order to convert parsecs into standard units like kilometers or meters, you must know the numerical value for the astronomical unit—it sets the scale for the rest of the universe. Its value was not know accurately until the early 20th century. In terms of light years, one parsec = 3.26 light years.

Using a parsec for the distance unit and an arc second for the angle, our simple angle formula above becomes extremely simple for measurements from Earth:

Parallax angles as small as 1/50 arc second can be measured from the surface of the Earth. This means distances from the ground can be determined for stars that are up to 50 parsecs away. If a star is further away than that, its parallax angle p is too small to measure and you have to use more indirect methods to determine its distance. Stars are about a parsec apart from each other on average, so the method of trigonometric parallax works for just a few thousand nearby stars. The Hipparcos mission greatly extended the database of trigonometric parallax distances by getting above the blurring effect of the atmosphere. It measured the parallaxes of 118,000 stars to an astonishing precision of 1/1000 arc second (about 20 times better than from the ground)! It measured the parallaxes of 1 million other stars to a precision of about 1/20 arc seconds. Selecting the Hipparcos link will take you to the Hipparcos homepage and the catalogs.

The actual stellar parallax triangles are much longer and skinnier than the ones typically shown in astronomy textbooks. They are so long and skinny that you do not need to worry about which distance you actually determine: the distance between the Sun and the star or the distance between the Earth and the star. Taking a look at the skinny star parallax triangle above and realizing that the triangle should be over 4,500 times longer (!), you can see that it does not make any significant difference which distance you want to talk about. If Pluto’s entire orbit was fit within a quarter (2.4 centimeters across), the nearest star would be 80 meters away! But if you are stubborn, consider these figures for the planet-Sun-star star parallax triangle setup above (where the planet-star side is the hypotenuse of the triangle):

the Sun — nearest star distance = 267,068.230220 AU = 1.2948 pc

the Earth–nearest star distance = 267,068.230222 AU = 1.2948 pc

Pluto–nearest star distance = 267,068.233146 AU = 1.2948 pc !

If you are super-picky, then yes, there is a slight difference but no one would complain if you ignored the difference. For the more general case of parallaxes observed from any planet, the distance to the star in parsecs d = ab/p, where p is the parallax in arc seconds, and ab is the distance between the planet and the Sun in AU.

Formula (1) relates the planet-Sun baseline distance to the size of parallax measured. Formula (2) shows how the star-Sun distance d depends on the planet-Sun baseline and the parallax. In the case of Earth observations, the planet-Sun distance ab = 1 A.U. so d = 1/p. From Earth you simply flip the parallax angle over to get the distance! (Parallax of 1/2 arc seconds means a distance of 2 parsecs, parallax of 1/10 arc seconds means a distance of 10 parsecs, etc.)

A nice visualization of the parallax effect is the Distances to Nearby Stars and Their Motions lab (link will appear in a new window) created for the University of Washington’s introductory astronomy course. With this java-based lab, you can adjust the inclination of the star to the planet orbit, change the distance to the star, change the size of the planet orbit, and even add in the effect of proper motion. (source : www.astronomynotes.com)

Units in Distance

  1. Astronomical Unit (A.U). It is defined as the mean distance of the Sun from the Earth. Its value is about 149,6 million km. This unit is conveniently used to express distance to the object in solar system because we can directly compared the distance to Earth-Sun distance.
  2. One light year is defined as the distance that light has traveled in light years. Light has velocity about 300.000 km/s. So, one light year equals to 9,46 x 10^12 km. This unit is mostly used to express the distance of extragalactic object. Remember that light’s speed is finite so distant objects are seen as they are in the past. For example the Sun. The Sun that we see at this moment is the Sun as it was 8 minutes ago. Light needs about 8 minutes to travel the Earth-Sun distance. So, looking farther objects mean we’re looking even further to the past. That’s why light years is more commonly used to express distant object’s distance. When we say that a cluster’s distance is 8 billion light years, it means that the cluster that we seen right now is the way it looks 8 billion years ago !
  3. Parsec (Parallax second). Star that have parallax 1 arc second have distance about 3,26 light years or 206.265 A.U (astronomical unit). Astronomer use this distance as a unit to express distance of the star. It is called a parsec. This unit is favorable to express star’s distance because it is closely related to star’s parallax (p). (remember that parallax = 1/distance, while the observer is on Earth, parallax is expressed in arc second and distance is expressed in parsec).

So, for reviewing our understanding about the parallax, try to answer these questions:

  1. If a star has parallax 0″,711, determine its distance (in light years) from us!
  2. Assume we can measure parallax from Mars (with the same technology that we used here on Earth). Assume that we can measure accurately using parallax method until 200 parsec from the Earth (distance limit). Determine the distance limit if we conduct the measurement of star’s distance using parallax method. Given that the distance of Mars from the Sun is about 1,52 AU.
  3. You observe an asteroid approaching the Earth. You have two observatories 3200 km apart, so you can measure the parallax shift of the incoming asteroid. You observe the parallax shift to be 0,022 degrees.Determine : (a) the parallax expressed in radians (b) the asteroid’s distance from Earth.
  4. If you measure the parallax of a star to be 0,1 arc seconds on Earth, how big would the parallax of the same star for an observer on Mars?
  5. If you measure the parallax of a star to be 0,5 arc seconds on Earth and an observer in a space station in the orbit around the Sun measures a parallax for the same star to be 1 arc seconds, how far is the space station from the Sun ?

You can share your solution of the above questions in the comment column.