Barnard’s Star

Introduction

Barnard's Star is a low-mass (0.16 solar mass) red dwarf star, which is located around 6 light-years away from Earth, in the constellation of Ophiuchus.

It is the second-closest star system to the Sun, and the fourth-closest individual star after the three components of the Alpha Centauri system (~ 4.4 light years away).

In 1916, the astronomer E.E. Barnard discovered that it had the largest known proper motion (apparent movement across the sky) of any star relative to the Sun. Its path will bring it

within 3.8 light-years of the Sun in 11,700 AD, when it will become our nearest neighbour.

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Introduction

This task will use images (or data) from the robotic Liverpool Telescope, in order to calculate the distance and transverse velocity (sideways to our line-of-sight) of Barnard’s Star.

The Liverpool Telescope (LT) is a professional robotic telescope located on the island of La Palma in the Canary Islands. It is built around a 2-metre diameter mirror, is approximately 8 metres tall and weighs in at around 25 tons. The site is chosen because the weather is good

most of the year, there is little light pollution and it is high up in the atmosphere.

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Workshop Resources

Before starting the task, make sure that you have access the LTImage image processing software, and have downloaded the following workshop resources:

All the image data that we will be analysing (21 files), and

The pre-prepared Barnard’s Star EXCEL worksheet.

The EXCEL worksheet allows you to input the results of all your measurements and outputs the distance and velocity values that we are looking for. Although there are some complex calculations being undertaken on the worksheet, most of them are hidden or can be ignored, allowing us to focus on the resulting motion graph and some few fairly simple measurements and calculations.

First, however, we need to talk about Parallax – this is a physical concept we experience every day of our lives, but don’t really tend to think about too much.

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Parallax Motion

“Parallax is an apparent displacement of an object viewed along two different lines of sight, and is measured by the angle or semi-angle of inclination between those two lines.”

In other words, if you move your head from side to side, closer objects will appear to move across more of your field-of-view than more distant objects. In the image above, a sideways step by the photographer results in the lamppost moving noticeably against the distant tower.

We can use this effect to measure the distance of a nearby star.

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Parallax Motion

As the Earth orbits the Sun, it has the same effect as the photographer stepping sideways, or as one moving their head from side-to-side whilst looking at their finger against a distant wall.

The angle the closer object subtends (traces out) is far greater for the nearby object than for a distant object. In astronomy, background objects tend to show no noticeable movement, so we

can use basic trigonometry to calculate the distance of nearby stars.

The result of the equation will be in multiples of the distance between the Earth and the Sun, a unit known as the Astronomical Unit or AU. Note that 1AU = 149,500,000 km.

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dp

1tan

Measuring Angles

In order to calculate the distance of a star, we first need to figure out a way of measuring the angle (p) traced out by the parallax motion. This is not as hard as one might think.

The Liverpool Telescope (LT) is designed in such a way that each pixel on the detector spans a known angle across the night sky of 0.264 arcseconds of angle. But what is an arcsecond?

Most people measure angles in degrees, however, astronomers tend to work with tiny angles and need to break degrees down into much smaller units. There are 60 arcminutes in a degree,

and a further 60 arcseconds in each arcminute, i.e. 3600 arcseconds in a degree.

An LT image is 1024 x 1024 pixels, or 4.5 x 4.5 arcminutes.

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A five pence piece at a distance of 3km, subtends a angle of 1

arcsecond. This equates to 4 pixels on a Liverpool Telescope image.

Taking Measurements

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We will now use observations from the LT over the past six years to measure Barnard’s Star in relation to two adjacent stars that are known to be fixed.

Star 1Star 2

Barnard’s Star

So how exactly will we do this?

In effect, we will be creating a graph with Star 1 at the origin, and the x-axis running through Star 2, both of which do not move.

All measurements of Barnard’s Star can now be plotted on a graph in relation to this new Star1/Star2 co-ordinate system.

x

y

Taking Measurements

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Although there is a little bit of complex mathematics in what we are about to do, you don’t really need to know much about it. All we need to concentrate

on is taking accurate measurements.

Star 1

Star 2

Barnard’s Star (BS)

So how will we do this?

Once the LTImage software is open, select an image to analyse. Make a note of the

image name, and then measure and record the following (pixel) distances:

(1) Distance from Star 1 and Star 2

(2) Distance from Star 2 and BS

(3) Distance from Star 1 and BS

Taking Measurements

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Note 1 : these are real observations from the Liverpool Telescope, so you may have to contend with the following issues:

(1)When you load each image into LTImage, you may have to ‘scale’ them so that the stars are more clearly visible. The demonstration videos on the LTImage page will show you how to achieve this.

http://www.schoolsobservatory.org.uk/astro/tels/ltimage

(2) Each image of Barnard’s Star may be shifted or rotated in comparison to the others. Take care when identifying the three stars that we are interested in. You may then wish to zoom in to get a better accuracy during measurement.

Note 2 : because stars 1 and 2 are “fixed”, i.e. not moving much in relation to each other, you can save some time by adopting a fixed distance for the Star 1 - Star 2 distance of 130 pixels.

Recording Measurements

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When taking readings, try to measure from exactly the centre of one star to the next. For better accuracy, make 3 measurements and calculate the average.

Once you have completed the form, take it to your teacher so that the results can be entered into a pre-prepared EXCEL worksheet that plots the motion of

Barnard’s Star on a graph and calculates its distance and velocity.

Star 1- Star 2 Star 2- BarnStar Star 1- BarnStar

1

2

3

Average

An alternative Method

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Although you can use the provided EXCEL sheet to plot the graph and calculate the distance and velocity of Barnard’s Star, you could opt to go through a similar process that uses tracing

paper to create a graph of stellar motion, with measurements taken using a ruler.

Take the first image and mark the centre of all three stars on the tracing paper. Place the tracing paper over subsequent images and line up stars 1 and 2 with the associated marks (made using the first image). Now plot a new point for Barnard’s star, which will have moved. You should

see a wiggly path appearing on the tracing paper as you add more and more points.

Image 1 Image 2Tracing Paper

An alternative Method

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When you have finished plotting all points, try to draw a line through the middle of your points and measure the maximum deviation (in pixels) to the left and right of the star’s path due to the

parallax motion. Also measure the distance (in pixels) between the first and last points.

Whichever method you use, the following two examples will prove useful in explaining how we go about calculating the distance and velocity of Barnard’s Star.

You can convert your ruler measurements to pixels by knowing that the distance between

Star1 and Star 2 remains fixed at 130 pixels.

You also need to know that the first and last observations are 2032 days apart.

Calculating Distance

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Q1. How far away is a star if it moves back and forward 30 pixels (either side of its line of travel) over the year?

A1. The first thing to realise is that over one year, the star in question will the move an angle of twice p, because the Earth will have passed on both sides of the Sun during its orbit.

Thus 2.p = 0.264 × 30 = 7.92 arcseconds, so

p = 7.92 ÷ 2 = 3.96 arcseconds or

3.96 ÷ 3600 = 0.0011 degrees

Using trigonometry,

d = 1 ÷ tan (0.0011) = 52,087 AU

In reality our closest star, Proxima Centauri, is around 268,000 AU from the Sun, which means the maximum pixel shift that we are likely to see due to parallax will be less than 5 pixels.

Calculating Velocity

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Q2. How fast is our star moving if it travels 40 wiggly pixels over two years?

A2. We already know that our star is 52,087 AU away from the wiggle amplitude. So all we need is the angular movement, or apparent motion, of Barnard’s star across the telescope image and we can calculate the total distance travelled (D) over the 2 years.

Using trigonometry:

D = tan (0.264 × 40 ÷ 3600) × 52,087 = 2.67 AU

Thus, Speed = 2.67 AU ÷ 2 years

= 1.335 AU/Year = 6.4 km/s

since there are roughly150,000,000 km in an AU and

365.25×24×3600 seconds in a year.

Checking your results

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And that is pretty much it, apart from making sure that the results you obtained are broadly in line with the official figures:

Distance : Astronomers have calculated that Barnard’s star is 5.98 light-years away, but given the inexact nature of our measurement process, you are doing well if you get within 2 light-years of that value.

Velocity : The official value is around 90 km/sec, but again, if you get within 20% of that figure, then you are doing very well. Don’t forget, this is only the velocity perpendicular (sideways) to our line of sight.

So, by combining simple measurements with some basic trigonometry, we are able to calculate the distance and velocity of a nearby star. Hopefully, this task highlights what a wonderful tool mathematics is for solving real-life puzzles.