Note: This is part of my old website content that I’m transferring here.
Question of pair alignments of quasars across galaxies is considered. It is assumed that the pair alignments are ideal bipolar ejections, i.e. that the pair is ejected simultaneously at the same velocity from exactly the opposite sides of the centering galaxy. A model is derived based on those assumptions, and known pair alignments are run through the model. Some statistical tests are performed. Most of the pairs don’t seem to be ideal bipolar ejections, but few pairs give good results. Possible application of the model for jets of galaxies and stars is noted.
Full work as PDF file:
Epilogue – A layman doing research
I went through a great deal of problems during this research project (the “Bipolar Ejections of Quasars from Galaxies” project). Here I will give a brief account of the project and the problems I encountered. I’m hoping that this might be helpful for someone in similar position than I was.
I started this project long time ago. I immediately encountered my first and rather surprising problem. I started sketching the situation of two quasars aligned across a galaxy on a paper, and with this I had a problem. Somehow I wasn’t able to do a sketch that would make sense. I kept pondering where the quasars should be placed when we look at them from Earth. Should they be placed to the celestial sphere having radius of the distance from Earth to the centering galaxy? Or, should they be placed on the line of our sight to the quasar and on the point nearest to the galaxy? I pondered these things for a while, and then I just put the problem aside.
Few monts later I decided to get back on the problem. I did another sketch and it was immediately the correct one (basically the same as the one presented in Fig. 1 of the paper). What I did differently this time was that now I considered the problem from the point of view of the galaxy and the quasars, not from the point of view of Earth. I just sketched a galaxy with quasars on both sides, nevermind where Earth was. Then I added Earth and immediately I saw where I went wrong before. Sometimes we just consider problems from wrong point of view and get stuck with it.
Next I started to model the situation mathematically. Not surprisingly, there were some problems with that too. Although I’m quite familiar with mathematics of the level used in this work, some equations turned out to be quite difficult to solve. So I tried many approaches and searched for information from Internet. Eventually I got them solved, but it took many days and many sheets of paper (yes, I still use mostly pen and paper for equation solving).
However, there is one mathematical problem I still haven’t solved, although I now think that it migh not be solvable. Originally I wanted to make the model as a sort of mathematical test. There was two sets of variables in the model; redshifts of the quasars and their angular distance from the galaxy. I wanted to calculate either ejection angle or velocities of quasars from redshifts and angular distances separately, and then compare the results. If the results match for some galaxy-quasar system, then that system is ideal bipolar ejection. But I just couldn’t get the two sets of equations to meet at any point, so eventually I abandoned that approach and started to build the model as it is presented in the document.
When the mathematical model was ready, I started to test it with imaginary cases of pairs of quasars ejected from central galaxy. The model was not giving right answers. I thought that the model was wrong and tried to find some errors from my derivation, unsuccesfully. The source of the problem turned out be the tool I was using to calculate the values, not the model. I had written my model to Microsoft Excel calculation sheet, anticipating the future calculations for many sample pair alignments, because repeating same calculations is very easy in Excel, you just copy your equations to the next row, input new values and that’s it. But as I repeated the calculations of imaginary cases carefully with a pocket calculator, and got the right answers, I realized that Excel doesn’t do the calculations with sufficient accuracy to be usable with my model. I started using mathematical software (Maple) instead of Excel, and there’s no problems like that anymore.
Now I was ready to start calculating the values for real pair alignments. So I searched scientific papers (mainly Arp’s) from NASA ADS for suggested pair alignments, digged up the data for the objects from NASA Extragalactic Database (NED) and run the values through my model. During this phase I had to learn how to calculate angular distance between two objects with known equatorial coordinates.
I thought my project was quite close to the finish (I even had started writing the paper), when another problem emerged. In an unrelated Internet discussion it was pointed to me (thank you, “Nereid”), that NED gives heliocentric redshifts for objects and that galactocentric redshifts (or even Local Group -centric redshifts) should be used when dealing with objects with large angular distances between them. Some of my pairs had angular distances of over ten degrees and I thought that I had to calculate everything again. But first I had to learn how to convert heliocentric redshifts to galactocentric redshifts. It was surprisingly difficult to find information on the conversion, but I finally found suitable formula from HyperLeda (thank you, David Russell, for pointing that way). I tried the conversion for few of my cases that I thought would be most affected. It turned out that the effect was so small that I thought I don’t have to do the calculations again. I felt relief, because each calculation takes few minutes and to do over 50 of them with the pace of couple of hours a week would have set me back many weeks.
Feeling of relief didn’t last long. It occurred to me that eventhough I had moved my model from Excel to Maple, I had been stupid enough to still use Excel for angular distance calculations. With cold sweat I checked how much difference there was when angular distances are calculated with Maple. There was enough difference to suggest that I should calculate everything again. I also decided that because I had to calculate everything again, I might as well do the galactocentric redshift conversion also. So I did.
After second round of calculations it was time to write the paper. It went well otherwise but it would have been much easier if I would know more about astronomy, my lack of knowledge shows especially in the conclusions and discussion section. You can see that I’m not actually saying much there except things that stay strictly within my ideal bipolar ejection model. I could be saying lot more if I would have more basic knowledge.
Overall this was very educational experience, and I can recommend it to other laypeople. If you have some idea what to research, just go for it.
April 24, 2006