### Basics of Astronomy: Photometry and allied stuff

Hello there!

If you are reading this post, I must assume you found it to do a background reading for the Summer school.

If you did not find this for the summer school, no probs! Just go ahead, and try to have a good read.

First off, the Summer school has been partitioned in 4 ways: (i). Introduction to General Relativity, Relativistic effects in Astronomy- almost purely qualitative,(ii). Electromagnetism, electromagnetic radiation in Astronomy, (iii). Basics of Astronomy- Optics, filters, and observations,  (iv). Machine learning and Deep learning in Astronomy.

Please note I will not put a lot of math, but instead refer you to corresponding books to do the same. Keeping in mind the session is for First year undergrads, I am trying to do some Mathematica coding for the same (if I am not lazy, that is )

Finally, don’t hesitate to post questions if any on the comments section!

Other lectures in this series:

If there is anyone interested to have  a discussion on any of these topics, you could comment here, of just search for the title on Quora, wherein I have uploaded all of this stuff under Abstracted Abstract Science.

IIT Madras boasts a lot of wonderful courses in various departments, but is painfully lacking in one field: Astronomy. There are courses which come close to Astrophysics, which would be Field theory and General relativity, but there are no basic courses in Astronomy. Hence, this post will be for introducing some basic aspects of astronomy, which you might have come across if you had prepared for the Olympiad.

Please note these concepts will be important while we discuss deep learning too. So keep this in the subconscious mind.

With this, let’s move on.

As we saw in the module on Electromagnetism, measurement of Electromagnetic quantities can be done in different wavelengths. Each wavelength gives some information on the material composition of area of observation. For example, it has been seen active, hot stuffs generally emit more in X-rays. UV rays, etc., while old, cold stuffs are generally in red, IR. There are a few more rules of thumb, however:

1. Dust absorbs shorter wavelengths and only lets longer wavelengths through.
2. An object may emit at short wavelengths, but it might get red-shifted so much, that the signal would appear at longer wavelengths.
3. Hotter bodies will emit at shorter wavelengths
4. Colder bodies emit at longer wavelengths. [These two are due to Wien’s displacement law discussed in Radiation]

These are all the frequency dependencies of astrophysical objects’ spectra. What about the intensity?

The intensity of light observed by an object is called its Luminosity. It is termed as Energy rate output from an object. But, we do not measure the actual energy emitted by the object, but rather, a fraction of that Luminosity falling onto a detector of a given area in a given bandwith. And we define this parameter to be the Brightness.

Brightness is used in the usual sense- meaning, how bright we perceive a given object to be. But then, we also need to account into the fact that the objects that we observe are not at the same distance- hence we introduce a distance measure in the terminology.

Apparent magnitude: It is termed as the Brightness of the object as observed by us, as observers. It is the brightness as seen through the naked eye. This quantity has a logarithamic definition, and hence brighter objects have lesser (or more negative) magnitudes, and dimmer objects have larger (or more positive) magnitude. For example, our Sun has an apparent magnitude of -27, while the comet 47P had a magnitude of +6 at its brightest.[1]

Absolute magnitude: The magnitude of light observed by us if the object were at 10 parsecs away from us. As simple as that.

If we know either of the magnitudes, and the Luminosity of the object, we can measure the distance of the object from Earth!

With the magnitudes, came in what was known as Hertzprung-Russel diagram. This plot is basically a Luminosity-Effective temperature plot of stars, which tries to group different kinds of stars together in a plot. There were some mathematical models done for Luminosity-Temperature relations, and these relations are approximately satisfied in the HR diagram, which is a measured, plotted, real-world data. The plot looks something like this:

Here, one can see that by plotting just the Luminosity and Temperature of an object, one can obtain the approximate age, and the kind of star observed. Neat right?

So far so good. But remember, we also have different wavelengths in which the measurement can be done? What about that? Which would be the best wavelengths to do measurements? And how would you define them?

For example, if one were to observe our good old Sun at different wavelengths, this is what one would see:

But wait! When we defined Apparent Magnitude as ‘how-bright-we-observe-stuff’, we did not think about how bright they would be in different wavelengths! For example, the Sun is bright in visible, but is almost dark in X-rays!

Houston, we have a problem!

Seeing this, some astronomers came to the rescue. They started defining different passbands, or a range of frequencies to be measured, and said:

Look, apparent and absolute magnitude still stand, but let’s take a set of Wavelengths, and look at what objects are bright and dark in this band. Also, as we have defined the wavelength range, we must also define what reference we take as our ‘0’ magnitude. For this, we choose a set of bright stars- stars which have proper Luminosity-spectral class defined, and take an average value for these bright stars, and take them to be our ‘Zero-value’.

The Zero-magnitude, after all computation, almost always falls down to the star Vega, which is assumed to be 0 in all given passbands. This work was done by Johnson and Morgan[4] for the Northern hemisphere, and expanded by Cousins[5] for the Southern Hemisphere. The filter functions defined by approximated by Bessel[6] (no, not the Bessel function guy). These functions look like:

So, one does measurements in each of these passbands, and uses these images for astronomy. For example, here’s an example of Crab Nebula in different passbands:

This is simply how ‘optical’ passbands are defined. However, I haven’t gone extensively in the definitions, and they can be found in the references[9].

Amateur astronomers, however have many such ‘filters’ for their convenience, and these are slightly different from the passbands defined above. Let’s have a look at them too[10]:

1. Solar filters: These are filters used to reduce the intensity of light from the Sun. You cannot simply point your scope at the Sun and observe it- unless you want your mirror/lenses to become gooey. We have a solar filter for our 8″ at the rooftop too.
2. Solar H-alpha filter: Hydrogen has a very prominent H-alpha line, and one can observe beautiful features and Supergranules on the Sun, not to mention Solar flares, Coronal mass ejections, and things like that. The solar H-alpha has an in built Solar filter (generally), with a cooler to maintain the center frequency and wavelength.
3. Polarizing filter: Basically your shades. A set of polaroids are used, and these can be used to change the intensity of light coming through the filter, to have multiple views at different intensities on the given object.
4. Nebular filter: O-III, H-alpha and H-beta filters are used for the corresponding band passes, to reduce light pollution and show in intricate features in the Nebulae. These are just filters, and are seldom accompanied by cooling devices. O-III and H-beta filters are combined sometimes into a broadband filter named UHC filter. All three filters are also combined into a broadband filter for eliminating Sodium and Mercury lines coming from light pollution.
5. Aberration filters: Chromatic aberration in refractors have special filters to be attached for correction.
6. Pseudo-UBVRI filters: These are a kind of UBVRI filters discussed previously, but they quite arent. They come in varities of Red, Yellow, Blue and Green (Basically, the first-generation Pokedex holders 😛 ). They are used mainly for Solar-system observations, enhancing and diminishing features like clouds, ice, atmosphere, etc.

Luminosity, it turns out, can also be used for distance-measurement in Astronomy. Distance measurements are not just made with luminosities, but there are a couple of techniques to do so. Let’s look at them one by one:

1. Parallax measurements: Remember your good old formula for ‘arc-length’ of a sector, which goes by $l=r\theta$ ? What if we just invert the equation, know the angle and the arc length, and find the distance of the object from Earth (or the radius in this case)? This is the basis of parallax measurement. One considers the arc-length to be the major-axis of Earth’s orbit, and does ‘parallax measurement’ and obtains the angle, thus obtaining the distance. However, note that this method’s resolution is limited by the Earth’s orbit’s major axis’ length, and hence can only be used for the nearby-stuff.

2. Standard candles: We have seen that Luminosity is the power output from a given body, while Brightness is how bright an object is as perceived by us. And we know that Brightness is Luminosity by the distance squared. With this inverse square law, and given our Filter equations (which was done qualitatively previously), we can determine the distance given we know the Luminosity of the object. But how do we get the luminosity? In this case, a fair bit of astrophysical modelling is involved. We use stars called Cepheid variables, which show a regular variation in their brightness, hence a defined Luminosity-Time period relation, and the Type-1a Supernovae, which have a definite Luminosity function. These are used for larger distance determination. (For example, all Type-1a Supernovae have same absolute magnitude.)
3. Main sequence fitting: If we know the metallicity and composition of a star, we can try and place it in a general area in the Main-sequence branch of the HR diagram (above). This will give us the absolute magnitude, and hence the distance.
4. Advanced methods: There are methods based on Baryonic oscillations, Globular cluster functions, and Supernovae themselves, but these are just too complex (yet) to be explained.

Wikipedia has an excellent article on this [11], and is summarized by this diagram:

I shall conclude this segment by just saying this: What I have touched upon is just the tip of the Iceberg. There are so many intricacies in here, which I have just averaged through. However, I provide an introductory list of references which will be a good start for this study.

References:

1. http://coolcosmos.ipac.caltech.edu/cosmic_classroom/cosmic_reference/luminosity.html
2. https://flipquiz.me/u/lcyp442/h-r-diagram-and-measuring-space
3. http://www.rxollc.com/technology/ni_optics.html
4. Fundamental stellar photometry for standards of spectral type on the revised system of the Yerkes spectral atlas: Johnson, Morgan
5. Revised zero points and UBV photometry of stars in the Harvard E and F regions: Cousins
6. UBVRI photometry. II – The Cousins VRI system, its temperature and absolute flux calibration, and relevance for two-dimensional photometry: Bessell
7. http://spiff.rit.edu/classes/phys373/lectures/colors/colors.html
8. http://qdl.scs-inc.us/2ndParty/Pages/16754.html
9. http://spiff.rit.edu/classes/phys440/lectures/filters/filters.html
10. https://www.wikiwand.com/en/Astronomical_filter