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Stellar parallax:
Space is big...really big. The distance between the Earth and Moon is hundreds of thousands of kilometers. The distance to the Sun is over a hundred million kilometers. But these are nothing compared to the distance between stars. For thousands of years, astronomers could tell that the stars were much, much more distant that any of the planets, but couldn't measure the distances.
The method to use was obvious: trigonometric parallax. Look very carefully at an object in the distance, from two different vantage points. As you move from one spot to the other, the target object will appear to shift position slightly, relative to more distant objects in the background. If you know the distance between the observation sites, and you can measure the angle by which the target shifts, you can use trigonometry to determine its distance. Scientists had tried for centuries, but had always failed because the angular shifts were too small to see.
Due to advances in telescope technology, their cleverness in picking stars which were likely to be nearby, and in part due to their persistence and hard work. In the 1830's Friedrich Bessel, Friedrich von Struve and Thomas Henderson were able to detect tiny angular shifts of three stars.
The diagram below shows how the parallax angle to a nearby star can be measured from measurements 6 months apart.
Once the parallax angle has been measured in arc-seconds, the distance to the star in parsecs can be calculated using the following equation.
USE OF THE EQUATION:
Friedrich Bessel measured the parallax angle of 61 Cygni to be 0.314 arcseconds. Use the equation above to calculate the distance to 61 Cygni in parsecs and then convert this to lightyears (1 parsec = 3.26 ly). Imagine what it may have been like in 1838 to discover that stars could be this far away from earth.
Follow this LINK to experiment with the simulation
Folder of Astro Problems
Limitations of parallax
From Earth, parallax angles less than 0.01 are difficult to measure. This means that the maximum possible distances that can be measured this way are about 100 pc. The distance to around 3000 stars have been measured from earth in this way. Recently two satellite missions have been able to measure more precise angles and therefore the distances to much more stars.
Hipparcos - A much better measurement
In 1989 the European Space Agency (ESA) launched a satellite to collect data about stars. This included measurments of the parallax angle to stars to down to a value of 0.002 arc-seconds. The satellite is called Hipparcos (HIgh Precision PARallax COllecting Satellite) in honour of the Greek astronomer Hipparchus. To date Hipparcos has measured parallax angles to nearly 120,000 stars.
Friedrich Bessel measured the parallax angle to 61 Cygni in 1838 to be 0.314 arc-seconds.
Modern Comparison:
Find the most up to date value for this angle from the Hipparcos catalogue as follows:
Click on this link.
Click on Browse this table... in the top left-hand corner.
Scroll down to the bottom of the page and enter the name of the 61 Cygni star into the box Object Name or Coordinates and press return. The Hipparcos code number to type in for 61 Cygni is HIP 104214.
Under the heading Hipparcos Main Catalog find the parallax angle for HIP 104214. The units of the angle as stated are [mas], which means milli arc-seconds. Compare this value to the Bessel value measured almost two centuries before.
Now calculate the distance to 61 Cygni using the Hipparcos value.
Gaia - The most advanced method to date:
The most recent satellite to measure parallax angle is Gaia. Launched by the European Space Agency (ESA) in 2013 it has been sending even more precise data to determine the distance to stars and is being used to create a 3D map of our galaxy - the Milky Way. Gaia can measure to a precision of 0.0001 arcseconds (10 microarcseconds). This represents distances of up to 100 kpc (the Milky Way is about 35 kpc in diameter).
The video below explains how Gaia uses the parallax method to measure distances. Lagrangian Points info.
Stellar Parallaxes - How to measure stellar distances with the Gaia satellite
The Centre for Astronomy at Heidelberg University has produced a 3D astronomy software application from the Gaia data called Gaia Sky. The Gaia Sky app can be downloaded here.
The Sky at Night (BBC) produced a programme in 2018 that focused on the Gaia satellite and how the data is being used. This is a 30 minute video. If you have time and are interested then watch this video in your own time.
The Sky at Night - Gaia: A Galactic Revolution
Measuring with Parallax - 4 Methods - Choose 2
Measuring with Parallax - 4 Methods - Choose 2
Method 1: The Parallax Method
Method 1: The Parallax Method
In the diagram below, P is a distance star which is used as a reference point. Q is a nearby star whose distance is to be calculated. The diagram on the right is more to scale and shows that if P is a very distant star then angle p tends towards zero and therefore the two lines AP and BP tend towards parallel. In reality AB is about one millionth of distance AP so this is a good approximation.
The Earth orbits the Sun once a year and so between June and December it is on opposite sides of the sun. The angle between P and Q is measured from A and B six months apart. This is angle a1 in June and a2 in December.
If AP and BP are parallel then they are also parallel to SQ so the angles a1 and a2 are the same as angles s1 and s2. The distance SQ (= d) can then be calculated using a simple triangle.
The Parallax Range Finder can be used to determine the distance to a nearby object in the laboratory.
Two points should be marked on the workbench 1 metre apart. This is the reference line.
The Range Finder should be placed on one of the marks with the two pins and centre line on the scale lined up.
Now turn the whole unit until the two pins are lined up with the distance object.
Then rotate the scale only until the two pins are lined up with the nearby object.
Read the angle from the scale.
Repeat this process with the Range Finder on the second mark at the other end of the reference line.
Add the two angles together and divide by two to give the angle s.
Method 2: Determining a real distance by triangulation.
Method 2: Determining a real distance by triangulation.
Determining a distance by triangulation
You need an angle-measuring tool (the professional one is called theodolite and is used by surveyors), one or two sticks, about as high as yourself and a tape measure.
Here's how we might measure the distance of a tree, like in the situation that is sketched above. Imagine you can walk from A until B, but you can't reach the tree, for instance because there is a river between you and the tree. The trick is that you determine the shape and the dimensions of the triangle ABC. When you measure AB (the baseline) and the two angles a (BAC) and b (ABC), the triangle ABC is completely known. As you may see, when the tree is more far away (C'), the angle at C becomes smaller and the shape of the triangle becomes different.
Mark two places, A and B, for instance by standing vertical sticks in the ground. The points A and B, together with the tree at point C, form a triangle. Measure the distance AB with the tape measure.
Method 3: Online Simulation (most realistic)
Method 3: Online Simulation (most realistic)
Method 4: Using a Camera
Method 4: Using a Camera
In this method the stars image's pixel distance from a reference point is measured. This method requires a nearby object of known length to act as a calibration for pixel distance. Photographs wil be taken from two different positions and LoggerPro used to determine the pixel distances.
1. Set up a star (ping-pong ball) and camera as shown below.
2. Find a reference point in the distance. For example, I chose a red window on the elementary school building.
3. Aim the camera at the star (try to make sure that the star is in the centre of the photograph) and take a photograph from one position on the track. Include a calibration object in the first photograph.
4a. Measure the length of the calibration object (pL) and
4b. Measure the distance of the calibration object from the camera (dL).
5. Move the camera along the track and take a second photograph with the star in the centre. Make sure that the reference object is still visible in the background. Record the distance (b) that the camera was moved along the track.
6. Upload the two photographs into LoggerPro by clicking on the Insert tab and selecting Picture => Picture with Photo Analysis.
7. Insert axes on the distant reference point to make it easier to measure the distance between the star and the distant reference object.
8. Determine the number of pixels between the star and the reference point as shown in the gif below. In this example the number of pixels is approximately 57.
9. Determine the distance between the star and the reference point on the other photograph.
10. Add the two distances together to get the total number of pixels (p) between the two star images.
11. Determine the number of pixels in the length of the calibration object (L) in the same way. Then use the equation above to calculate the distance to the star. 12. Measure the actual distance (using a method of your choice) and compare it with the calculated value. What is the percentage uncertainty in the measurement?
13. From the geometry of similar triangles the distance (d) to the star can be calculated from the following equation:
where
L = actual length of the calibration object
b = actual distance the camera was moved
dL = actual distance of the calibration object from the camera baseline
p = distance as the number of pixels between the star images
pL = length as the number of pixels of the image of the calibration object
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