## Differential Equations -A

Posted March 25, 2010 By grigoris

Solutions of a second order differential linear and homogeneous equation with constant coefficients:

a y” + b y’ + c y = 0 (a ≠ 0)

Solve the characteristic equation: a r2 + b r + c = 0
with general solutions r1,2=[(-b±√b2-4ac )/2a].

1> if Δ=b2-4ac > 0 then the solutions r1=r1 are real and
the general solution of the differential equation is

y = c1 er1x + c2 er2x

2> if Δ=b2-4ac = 0 then the solutions are real and equal (r1=r>2) and the general solution of the differential equation is

y = c1 er1x + c2 x er1x

3> if Δ=b2-4ac < 0 then the solutions are complex and the general solution of the differential equation is

y = c1 ekxcos(lx) + c2 ekxsin(lx)

where r1,2=k±il=(-b/2a)±i[√b2-4ac /2a]

## Manipulating postscript/pdf files

Posted March 24, 2010 By grigoris

Postscript files (.ps) are usual and in many cases some manipulation is needed. [for more see also here]

Convert from a ps to pdf file:
`\$ ps2pdf input_file.ps output_file.ps `

Merge 2 or more files:
`\$ psmerge -ooutput_file.ps input_file_1.ps input_file_2.ps [more_files]`
note: that -o and output_file’s name are not separated by any space

Put more pages per page:
`\$ psnup [options] -n input_file.ps output_file.ps`
[options]: -l , -r which “corrects” for orientation
-n: number of pages from input_file.ps that will be on a single page of output_file.ps (for example -2 for 2 pages/new page, -4 for 4, etc.)

## Sexagesimal and radian to degree conversion

Posted March 23, 2010 By grigoris

A. If you want to convert sexagesimal numbers (RA=hh:mm:ss.s Dec=±dd:mm:ss) to degrees then use these relations:

RA>deg: 15 x (hh + mm/60 + ss.s/3600)
Dec>deg: ±dd ± (mm/60 + ss/3600)

These relations are pretty easy to remember since
> 1 hour the sky rotates 15 degrees (360degrees in 24 hours)
> 1 minute = 60 seconds
> 1 hour = 60 minutes = 60 seconds
> Caution with Dec: when Dec is positive you add the rest while when it is negative you subtract.

B. For radians to degrees conversion use the relation:

## WhereIsM13 and .jar files

Posted March 18, 2010 By grigoris

WhereIsM13 is a great 3d visualization application of deep sky objects! You can see where these objects lie in our Galaxy along with their physical properties.

For linux users download the .zip version which includes a .jar file that can be run by double clicking (well, not in my case…although I changed the properties so as to be executable). If not, then you can easilly run it by typing:

`\$ java -jar /path/to/file.jar`

(there is no need for the file to be executable!).

## William Parsons – first direct observation of spirals

Posted March 18, 2010 By grigoris

William Parsons, was the first to notice spirals in a galaxy. His first scetch was the M51 (Whirlpool Galaxy) in 1845.

More on William Parsons:
> Wikipedia
> A list of photos & discoveries
> Drawings

## DSLR processing and photometry for epsilon Aurigae with IRIS

Posted March 17, 2010 By grigoris

Prepared for the CitizenSky.org ‘s DSLR tutorials (with the help of bikeman and Roger_Pieri). CitizenSky.org is dedicated to the eclipsing binary epsilon Aurigae, but the procedure is more general and can be applied to any star.

~ ~ ~

For easiest use download the last version of IRIS 5.58 (especially for any Linux user because the new command convertraw is essential to load raw images). Then you can install it very easily either to Windows or Linux (using wine).

[Note1: for astronomical images select the Linear interpolation mode for the RAW decoding toward a color images (less artifact around pointlike objects).

Note2: when you will be loading series of images the IRIS will pause for some time to read, trasform,calculate and save the images (depending on the number and the computer of course). Only the final output of each procedure will be visible but that does not mean that IRIS has rejected the rest (you specify a working directory and the program will look into this according to the input you give).

At the final image that is loaded in IRIS you may not be able to see something. That does not mean that nothing exists! You can increase the visibility of the stars by moving the upper slider in the Threshold Window to the left until stars appear with enough contrast. This is just changing the display settings, not the image itself. In most cases the AUTO selection works fine. In the whole processing procedure you don’t really care to look these images as all these are intermediate steps to make the final image that we are looking for (step 6a).

Step 1 – Initialization:

1a. Open File>Settings and choose your working path (where you have COPIES of your images), the script path (only if you have any scripts that you want to use, otherwise not necessary, the file type (PIC for DSLR RAW) [IRIS will convert raw images to CFA images].

[Note – different raw interpolation methods explained (linear, median, gradient):

For that the software must interpolate information collected by close pixels covered either with red filters, green filters and blue filters. The operation is so called demosaicing. The result is an image where each color plan is coded on 16 bits (either 48 bits of information by pixel). The bilinear method for interpolation is fastest, but it is passable concerning spatial resolution (restitution of the finest details). However this method remains often viable in many situation because the resolution is degraded at least same value at the time of the registration of a sequence of images. The median method preserves the resolution, but applies a filter which degrades the natural aspect of the stars for deep-sky imagery. This technique is not recommended for the stars images. The gradient method preserves well the resolution. The calculating time is long, but it is the method which gives the best results for the details point of view, even if in deep sky imagery the stars are not free from artifact.]

1b. Open Camera Settings (the button with the campera picture) and select only the right camera model you used and raw interpolation method linear [DO NOT tick the Apply button of White Balance]. The rest are not needed (leave the default values).

2a. Click Digital photo > Decode RAW files > (opens a window where you drag and drop your files) and drag & drop your DSLR raw images (e.g. for Canon 300D: *.CR2 files).

The dialog lets you specify a sequence name. then you press “–>CFA” to decode the images.

You do this repeatedly for:

star field images (e.g. use name ‘img’)

dark frames ( suggested name ‘dark’)

flat field images ( suggested name ‘flat’)

bias frames ( suggested name ‘bias’)

After this you will have files named

img1,img2,…

dark1,dark2,…

flat1,flat2,….

bias1,bias2,…

in the working directory configured in step 1a.

2aa. [alternative way – needed for linux users] Open the command line (button right left from the camera button) and type :

`>convertraw input_sequence output_sequence number`

input_sequence are the images from your camera and

output_sequence the images that IRIS created (in CFA format).

Do it for all the images that you have (star-field images, dark, flat, bias).

Step 3 – Preprocessing > Bias, Dark, Flat:

3a. Go to Digital Photo Tab and select Make Offset (Bias frame). Type the appropriate sequence name (the one used in step 2, e.g. ‘bias’) and the number of bias images. Then select the command line button (the one on the left from camera button) and type:

`>save master-bias`

of the created image (you can use of course whatever name you want!)). You can also save it by File>Save , but the command line can be kept opened all the time as it will be easier to use.

3aa. IRIS needs a master bias frame (also called offset) to continue processing. So in case you don’t have bias images you can create a “fake” one. Open a star field image (or one from the sequence img) and type at command line:

```>fill 0 >save master-bias```

3b. Do the same for dark, you enter the appropriate generic name (only the sequence name), the offset image name you created previously (‘master-bias’), the number of dark frames and the method median (median or mean should both work well). Also, go and save the image as ‘master-dark’ (either type: save master-dark , or File>Save).

3c. Improve dark frame substraction: open the master dark frame (the image that you created at step 3b, which should be already opened) and at command line type:

`>find_hot cosme 500`

and check the output box (should open automatically). There is a value which should be 100<output<200.

3d. At the same Tab (Digital Photo) do the same for flat-fields and select Normalization value = 1000. [After creating the image you can check the output of the command STAT and see what is the max output value. This should be less that 32767. If not arrange accordingly the normalization value and check again. You just have to be under 32767].

Before saving the final flat image type at command line:

`>grey_flat`

so as to normalize the CFA-flat and then save the flat field to “master-flat” (either type: save master-flat , or File>Save — by now you should have understood why the command line is faster and easier !!)

3dd. In case you do not have flat images you can create a flat sequence as follows: first select 3 images from the star field images (prefferable the first, the median and the last one – as the camera is not moving the field rotates so by using these images you can remove the stars and take correction for system, lens+camera, of course this should not substitute entirely the processing for making flat images ! But in some cases…it can be really handy.). RENAME them as flat1 flat2 and flat3 (let’s call this sequence flat). In order to make the final master-flat image type at command line:

`>sub2 flat master-bias i 0 number`

(choose a region 100×30 pixels or more in the image – does not matter where exactly)

```>opt2 i master-dark i number >ngain2 i 1000 i number >smedian i number >grey_flat >save master-flat```

where flat is the sequence we created, master-bias (or offset or anything you named the saved image of step 3a), master-dark (step 3b) number the number of images (in our case 3). i is a sequence that helps us to build the master-flat. Also in this case you can check first with STAT command if the output after SMEDIAN is correct (according to what is written in step 3d) and then procceed with the grey_flat (see step3d) and saving the image.

3e. Go to Digital Photo>Preprocessing. Input the sequence name of you data images (img) along with the offset map (master-bias, step 3a), dark map (master-dark, step 3b), flat-field map (master-flat, step 3d), cosmetic file (step 3c). Give an output sequence name (e.g. ‘img-cal’) and the nunber of files. It is not necessary to tick dark optimize as it will take a little bit longer without real effect on the final result [of course you can use it if you want !]

Step 4 – Alignment of Images:

4a. Go to tab Digital photo>sequence CFA conversion and select the sequenace name that you gave at step 3e (‘img-cal’). Give an output name (‘img-cal-conv’) and number of your star images and select Color output files type .

4b. Go to Processing tab>Stellar registration and put the sequence name of 4a (‘img-cal-conv’), give output sequence name (e.g. ‘img-reg’) and number. Choose Global matching and Quadratic transformation

4c. Stacking of images: Processing tab>Add a sequence and give sequence name of 4b (img-reg) along with the number of images and method arithmetic (select also “normalization if overflow if not selected).

4d. Save the image!

Step 5 – Selecting green channel:

5a. Go to Digital Photo Tab > RGB Separation (so as to take only the green channel in which we are interested) and enter names for the color channel files, e.g.

final-r

final-g

final-b

You should be able to see see these images at your working directory.

Note: before performing the rgb separation take a look at the image (by selecting AUTO of Threshold Window). You can actually see the colours of the stars !! Then for scientifc reasons we select only the green channel and the images become less beautiful (but more interesting!).

Step 6 – Photometry:

6a. IRIS opens an image with a different way than some other programs. So load (either by File > Load or at command line by typing >load final-g ) the final-g image (from step 5a) and take a carefull look to figure out your field (it should not be that hard for eps aur!).

You can increase the visibility of the stars by moving the upper slider in the Threshold Window to the left until stars appear with enough contrast. This is just changing the display settings, not the image itself. In most cases the AUTO selection works fine.

6b. Go to Analysis>Aperture photometry. With our sample data, you can probably use the default values for aperture, but for your own data you may need to change these sizes so you may wish to review how to size an aperture . If you place the circles to any object you will take an output at the output box with values for Intensity and Magnitude (along with other parameters). The important one is the Magnitude (or Intensity which are equivalent).

[This section adapted from citizensky’s DSLR documentation & reduction team:

> The inner circle defines the area where the star has to fit in. The pixels inside this area must contain all (or at least almost all) light from the star. It should contain a bit of extra sky but never a second bright star.

> The outer ring between the middle and the outer circle defines the area that is considered to be “sky background”. As a rule of thumb, the radius of the outer ring should be about the same as the radius of the inner circle, but on the other hand it should not be so big that stars near any of the stars that you measure will be included in this ring.

> The area between the outer ring and the inner circle is just there to separate the two areas, pixels in this area are ignored. You will set one aperture to fit all stars (variable and comparison stars) that you will measure in an image.

As long as you are working with the same camera, lens, and focus setting, you need to make this decision only once and use the setting all over again for your measurements. When doing the measuremts, you use the aperture like a reticule to “capture” the light of the star you want to measure.]

6c. Now, locate all of the comparison stars you wish to use (see the table of comparison stars). If you have already activated the photometry tool (already done at step 6b) you should see the circles at the mouse point and a tick -activated- left to Aperture photometry option at the Analysis tab). Now, check your comparison star (i.e. eta Aur) and record the output in the spreadsheet or on paper to three decimal places. Repeat for as many check stars you want to use. Finally you select epsilon Aurigae to take an instrumental value of its magnitude.

6d. Repeat step 6c for all your sets of images. As long as you are using the same lens and DSLR, you don’t have to re-create a master-flat and master-bias for every observation session! You can save them for re-use and then skip the steps 3a and 3d.

6e. Use the Excel Data Reduction spreadsheet that is available to extract the final results.

## Interesting Hertzsprung–Russell diagrams

Posted March 8, 2010 By grigoris

Two very interesting images about the H-R diagram!

The Hertzsprung–Russell diagram is a scatter graph of stars showing the relationship between the stars’ absolute magnitudes or luminosity versus their spectral types or classifications and effective temperatures.

This diagram below is a plot of 22000 stars from the Hipparcos Catalogue together with 1000 low-luminosity stars (red and white dwarfs) from the Gliese Catalogue of Nearby Stars.

## Software for variable stars

Posted March 6, 2010 By grigoris

A not exhaustive list of software suitable for variable stars (e.g. reduction of images, photometry, period search, etc.).

1. IRIS

2. Period04

3. IRAF

4. AIP4WIN

5. Mira

7. MaxIm DL

8. Peranso

10. VStar on CitizenSky.org, AAVSO, or Sourceforge

11. AVE