Astr211

Assignment 4

Jacob Darby – 36443458

13/ 08/ 2015

Figure one: Images 1-5 showing the Asteroid travel across the first star field.

Using Astronomical program ds9 to identify the asteroid:

Ds9 is used instead of AIP4WinV2 as ds9 contains a function which allows you to blink multiple images unlike “AIP4WinV2” which only allows for two. After opening the ds9 program each image was opened in different frames. The nine images opened where taken over a 20 minute interval by Pam Kilmarten and Alan Gilmore using the 1.0-m telescope at our observatory at Mt John, in which the telescope changed position slightly for the last four images. Thus after uploading all the images on separate frames these frames where then match up using the tool Frame > Match Frames > WCS

which eliminated this position change and progression of time over the images. Following this the image black/white scale was set “92.5%” for the first 5 images and “99%” in order to achieve a

clearer set of images in which there brightest objects are illuminated. With a set of nine clear images the blink tool was used Frame > Blink Frames to iterate through the images and thus the fast moving

asteroid could be clearly identified moving across the screen. As seen in juxtaposing the following first five images the asteroid is traveling through the star field as time passes.

Making plots of the position change in the asteroid:

For tracking the position of the asteroid the astronomical program AIP4WinV2 was used. A report and calibration of each image must be completed in order to track the overall movement of the asteroid across the two star fields. To begin all images where loaded onto “AIP4WinV2” and the

astronomic calibration process was begun changing the pixel dimensions under the imager in the image display control which were redefined as “15 by 15 micrometres”. Note that all images

Black/White setting was altered to obtain a much clear view of the star fields the white was changed from over 2000 to “680” for all images to achieve this. Following this the Astrometry tool was where the reference data was loaded “overlay1.gsc” for the first five images and “overlay2.gsc” for the final four as we are referencing two different star fields in which the asteroid has travelled through. Once this was loaded and after the search radius was set to five pixels six references stars where selected R1 to R6 whom were the brightest and largest stars ranging the entire star field making them easily identifiable. Then finally the target T1 was selected and a report on that image was generated. This

process was repeated for all nine images across the two star fields and thus nine reports where generated. Images of the process can be seen below in “figure two” and “Figure three”.

Figure two: Astronomic calibration shown for images 4 and 5 in star field one

Figure two: Astronomic calibration shown for images 6 and 7 in star field two

For Every image during this process the focal length and position angle where recorded as seen in “Table one”.

Image Number Focal Length(mm) Position Angle(degrees)One 2399.27 -0.7654Two 2398.62 -0.7915

Three 2397.49 -0.7639Four 2400.17 -0.8087Five 2398.20 -0.7760Six 2403.78 -0.8472

Seven 2401.30 -0.8314Eight 2400.30 -0.8874Nine 2401.16 -0.8773

Table one: Table showing the focal length on the telescope in mm and its position angle in degrees to match up to the reference data.

Calculation of the Proper motion of the fast moving Asteroid:

Following the calibration and report generation process the right ascension and declaration values from the reports were analysed as these will aid in the calculation of the proper motion of the

asteroid. These two values represent the two coordinates that astronomers use to describe the position of celestial objects. Where declaration is a measure in degrees north or south of the celestial equator an equator which runs through the same plane as the earths equator and whose poles are the same as the earths. Furthermore the right ascension which is measured in hours, starting at a

point where the ecliptic – the apparent path of the sun round the earth – crosses the celestial equator each March.

The following relevant information was taken from the generated reports in “Table two”.

Image Report RA(hrs:mins:secs) DEC(degrees:arcmins:arcsecs) Time of obs(hrs:mins:secs)One 09:06:55.68 -52:58:17.8 09:11:30.350Two 09:07:10.44 -52:58:07.3 09:11:55.250

Three 09:07:25.32 -52:57:56.4 09:12:20.350Four 09:07:39.96 -52:57:45.6 09:12:45.350Five 09:07:57.84 -52:57:32.1 09:13:15.350Six 09:19:26.81 -52:46:31.9 09:31:35.250

Seven 09:19:46.58 -52:46:08.2 09:32:05.250Eight 09:20:03.08 -52:45:48.2 09:32:30.250Nine 09:20:19.59 -52:45:28.1 09:32:55.250

Table two: Table showing relevant data extracting from the generated reports which can be used to calculate the proper time of the asteroid

Following this to calculate the proper motion of the star over its entire motion across two star fields (nine images) we must find the proper motion vector between each image. From this we can see how

the direction and magnitude of the proper motion changes over its entire imaged path and make a conclusion on whether is path linear or has a curved trajectory. For this to be completed we must

convert both our change in right ascension and declination into units of arc-seconds and the find the rate of change of the two between images through the following calculation:

Image one and two will be used as an exemplar for all other images as the same method can be implemented between all images.

First must calculate the change in right ascension and declination.

∆ RA=RA2−RA1

(Where RA1is the right ascension in image one)

∆ DEC=DEC2−DEC1

(Where DEC1 is the declination in image one)

∆ RA=¿09hrs 07mins 10.44secs – 09hrs 06mins 55.68secs = 0hrs 0mins 14.76secs

∆ RA=¿14.76secs

∆ DEC=¿ -52 degrees 58arcmins 07.3arcsecs – (-52 degrees 58arcmins 17.8arcsecs)

∆ DEC=¿ -10.4arcsecs

Then in using “table three” these values can be converted into degrees and then arc-seconds.

Unit Value Sexagesimal system SymbolHour 1

24circle

15 ̊� (H)

Minute 160

hour, 1

1440circle

(1/4) , 15’ ̊� (m)

Second 160

minute, 1

3600hour,

186400

circle(1/240) ,(1/4)’, 15’’ ̊� (s)

Table three: Table showing the astronomical conversions between relevant units.

Therefor from this table we get can convert right ascension to the following

∆ RA=14.76 seconds=14.76×15=221.4 ' '

∆ RA=221.4 ' '

∆ DEC=−10.4 ' '

From these values in order to calculate the proper motion of the asteroid between images one and two we must find both the rate of change in both right ascension and declination and the us simple

Pythagoras to find the proper motion of the asteroid between the two images.

∆ t 12=t 2−t1

(Where ∆ t 12 is the time between when image 1 and 2 were taken this can be found from table two)

∆ t 12=¿09hrs 11mins 55.250secs – 09hrs 11mins 30.350secs = 25.0 seconds

∆ t 12=25.0 seco nds

Therefor now the rate of change of both declination and right ascension can be found

μα=∆ RA∆ t 12

= ∆α∆t 12

=221.4 ' '25 secs

=8.856' ' /second

μα=8.856' '/ second

μ∂=∆DEC∆ t12

= ∆∂∆ t 12

=−10.4 ' '25 secs

=−0.416' ' /second

μ∂=−0.416' ' /second

From this and finally finding the average declination angle between the two images proper motion can be found.

DEC Av=DEC1+DEC2

2

Where “DEC1” and “DEC2” are in units of degrees thus

DEC1=−52degrees58' 17.8' '

1'=( 160 )degrees

1' '=( 13600 )degrees

Therefor from the follow conversions

58'=(5860 )=0.967degrees

17.8' '=( 17.83600 )=4.94 ×10−3degrees

DEC1=−52−0.967−4.94×10−3=−52.962degrees

DEC1=−52.962degrees

Same calculation was made for DEC2 as well and obtained a value of

DEC2=−52.969degrees

DEC Av=−52.962−52.969

2=−52.966degrees

DEC Av=−52.966 degrees

Now we have all the relevant data to find the proper motion vector of the asteroid between images one and two using the following right hand triangle.

(Where “μ” is the proper motion value and “∂” is” DEC Av”)

μ2=μ∂2+μα

2 cos2(∂)

Therefor the magnitude for the proper motion of the asteroid between images one and two is

μ2=(−0.416)2+(8.856)2cos2 (−52.966)

propermotion=μ=28.623

Thus the direction angle is the top left hand angle in the triangle above.

tan∅=μ∂

μα cos (∂)=−0.416

5.334=−0.0470

∅=−2.689degrees

This process is repeated for all images and the proper motion of the asteroid is found for each set of images as seen in the table below.

Image sets

∆ RA(‘’)±0.05 ' '

∆ DEC(‘’)

±0.05 ' '

∆ t (secs) μα

±0.05μ∂

±0.05∂(DEC Av)( �̊ )

±0.1 ' 'μ(proper motion)±0.3

∅ ( �̊ )

1-2 221.4 -10.4 25.0 8.856 -0.416 -52.966 28.623 -4.460

2-3 223.2 -10.9 25.1 8.892 -0.434 -52.967 28.869 -4.633

3-4 219.6 -10.8 25.0 8.784 -0.432 -52.964 28.179 -4.668

4-5 268.2 -13.5 30.0 8.940 -0.450 -52.961 29.202 -4.777

5-6 10334.55 -660.2 1099.9 9.396 -0.600 -52.867 32.532 -6.038

6-7 296.55 -23.7 30.0 9.885 -0.790 -52.772 36.388 -7.525

7-8 258.3 -20.0 25.0 10.332 -0.800 -52.766 39.723 -7.282

8-9 247.65 -20.1 25.0 9.906 -0.801 -52.761 36.576 -7.611

Table Four: Table showing relevant data to calculate the proper motion vector for each set of images.

From this data the following graph was able to be plotted

As seen from the graph above

we can conclude that

the asteroid takes a linear path through the star field as time passes by. The magnitude of the proper

motion vector is seen to linearly increase also with a change in time from “8.856” to a value of “10.332”.

Comment on uncertainty values in calculation:

There was a initial uncertainty in the measurement of the right ascension and the declination of the asteroid in the images. This uncertainty arose from the fact that a manual click was required to mark

the location of the asteroid from image to image this marker may not have always been located at the centre of the asteroid and thus its actual positioning wouldn’t have been consistent from image

to image. From this the uncertainty of the right ascension and declination values were chosen as half the smallest value for both of these to account for this possible error. Thus the uncertainty was

±0.05 '' for both.

Therefor when working out the ∂(DEC Av) value the uncertainty value was now “ ±0.1 ' ' “due to the additive uncertainty rule

(A ± ∆A) + (B ± ∆B) = (A + B) ± (∆A + ∆B)

Following this the uncertainty in the magnitude of the proper motion was found in the squaring and the adding of three different uncertainties as seen bellow

uncertainty∈the propermotion=unc (μ)

unc (μ )=unc (μ∂2 )+unc ( μα

2)×unc (DEC Av)

Thus to calculate this uncertainty we must establish two more rules of uncertainties the power and multiplication rules

Where the multiplication rule works as follows

0 200 400 600 800 1000 1200 1400

-8

-7

-6

-5

-4

-3

-2

-1

0

Graph showing the path of a asteroid traveling through a star feild

Time (seconds)Angl

ular

dire

ction

of t

he A

ster

oid

(deg

rees

)

The Absolute Uncertainty = ∆m

The Relative Uncertainty = εm

(A ± ε A) x (B ±εB) = (A x B) ± (ε A+ εB)

And the power rule as follows

(A±εA)n = (An ± nε A)

There for in implementing both of these rules the following uncertainty for the magnitude of the proper motion was found as

unc (μ )=±0.3

Finally the uncertainty of the proper motions direction was found which is also shown on the graph

unc (∅ )=±0.6 �̊