# How to photograph shock waves ?

fyfd:

This week NASA released the first-ever image of shock waves interacting between two supersonic aircraft. It’s a stunning effort, requiring a cutting-edge version of a century-old photographic technique and perfect coordination between three airplanes – the two supersonic Air Force T-38s and the NASA B-200 King Air that captured the image. The T-38s are flying in formation, roughly 30 ft apart, and the interaction of their shock waves is distinctly visible. The otherwise straight lines curve sharply near their intersections.

Fully capturing this kind of behavior in ground-based tests or in computer simulation is incredibly difficult, and engineers will no doubt be studying and comparing every one of these images with those smaller-scale counterparts. NASA developed this system as part of their ongoing project for commercial supersonic technologies. (Image credit: NASA Armstrong; submitted by multiple readers)

## How do these images get captured?

It may not obvious as to how this image was generated because if you have heard about Schlieren imaging what you have in your head is a setup that looks something like:

But how does Schelerin photography scale up to capturing moving objects in the sky?

## Heat Haze

When viewing objects through the exhaust gases emanating from the nozzle of aircrafts, one can observe the image to be distorted.

Hot air is less dense than cold air.

And this creates a gradient in the refractive index of the air

Light gets bent/distorted

## Method-01 : BOSCO ( Background-Oriented Schlieren using Celestial Objects )

You make the aircraft whose shock-wave that you would like to analyze pass across the sun in the sky.

You place a hydrogen alpha filter on your ground based telescope and observe this:

Notice the ripples that pass through the sunspots

The different air density caused by the aircraft bends the specific wavelength of light from the sun. This allows us to see the density gradient like the case of our heat wave above.

We can now calculate how far each “speckle” on the sun moved, and that gives us the following Schlieren image.

## Method-02: Airborne Background Oriented Schlieren Technique

In the previous technique how far each speckle of the sun moved was used for imaging. BUT you can also use any textured background pattern in general.

An aircraft with camera flies above the flight like so:

The patterned ground now plays the role of the sun. Some versions of textures that are commonly are:

The difficulty in this method is the Image processing that follows after the images have been taken.

And one of the main reasons why the image that NASA has released is spectacular because NASA seems to have nailed the underlying processing involved.

Have a great day!

* More on Heat hazes

** More on BOSCO

*** Images from the following paper : Airborne Application of the Background Oriented Schlieren Technique to a Helicopter in Forward Flight

**** This post obviously oversimplifies the technique. A lot of research goes into the processing of these images. But the motive of the post was to give you an idea of the method used to capture the image, the underlying science goes much deeper than this post.

# Jackson’s Laplacian in spherical Coordinates

If you took a look at one of the previous posts on how to remember the Laplacian in different forms by using a metric,  you will notice that the form of  the Laplacian that we get is:

$\nabla^2 \psi = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial \psi}{\partial r} \right) + \frac{1}{r^2 \sin(\theta)} \frac{\partial}{\partial \theta} \left( sin(\theta) \frac{\partial \psi}{\partial \theta} \right) + \frac{1}{r^2 \sin^2(\theta)} \frac{\partial^2 \psi}{\partial \phi^2}$

But in Jackson’s Classical Electrodynamics, III edition he notes the following:

This is an interesting form of the Laplacian that perhaps not everyone has encountered. This can obtained from the known form by making the substitution $u = r \psi$ and simplifying. The steps to which have been outlined below:

# A note on the Hydrogen spectrum

The emission spectrum of atomic hydrogen is given by this amazing spectral series diagram:

Let’s take a closer look at only the visible portion of the spectrum i.e the Balmer series.

If a hydrogen lamp and a diffraction grating just happen to be with you, you can take a look at the hydrogen lamp through the diffraction grating, these lines are what you would see:

Source

These are known emission lines and they occur when the hydrogen atoms in the lamp return to a state of lower energy from an excited energy state.

Representation of emission and absorption using the Bohr’s model

Here’s another scenario that could also happen:

You have a bright source of light with a continuous spectrum and in between the source and the screen, you introduce a gas (here, sodium)

Source: Harvard Natural sciences

The gas absorbs light at particular frequencies and therefore we get dark lines in the spectrum.

This is known as absorption line. The following diagram summarizes what was told thus-far in a single image:

The absorption and emission spectrum for hydrogen look like so :

## Stars and Hydrogen

One of the comments from the previous post was to show raw spectrum data of what was being presented to get a better visual aid.

Therefore,the following spectrum is a spectrum of a star taken from the Sloan Digital Sky Survey (SDSS)

Plot of wavelength vs median-flux

Here’s the spectrum with all the absorption lines labelled:

Source: SDSS

You can clearly see the Balmer series of hydrogen beautifully encoded in this spectrum that was taken from a star that is light-years away.

And astronomers learn to grow and love these lines and identify them immediately in any spectrum, for they give you crucial information about the nature of the star, its age, its composition and so much more.

Source: xkcd

Have a great day!

*If you squint your eyes a bit more you can find other absorption lines from other atoms embedded in the spectrum as well.

# On the strong 5577Å spectrum line

The above is a plot of the Wavelength(in Å) in the x-axis vs the flux of some objects from the Sloan digital survey ( consists of galaxies, young stars, Quasars, etc)

But  there is one strong peak in all of those plots that seems to stand out: 5577 Å

And if you like, the color that it represents is the above  (Made with Stanford’s color matcher app)

## A nightmare for the astronomer

This line at 5577.338 is what astronomers refer to as a ‘skyline emission’ or a ‘mesospheric night-glow’ and arises from the recombination of atomic oxygen in the mesosphere.[2]

This line is of no significance to an astronomer who is looking to find out properties about a far away astronomical object. Yet, this line pops up in every spectrum of any object that you look at in outerspace!

In addition since the line is so strong, it contaminates the nearby pixels making the nearby data unusable and also messes up the scaling of the plot.

## Wavelength Calibration

What do you do with something that is always there but has no use for you? – Re-purpose it!

Having noticed that this peak was consistent at 5577.338, Astronomers decided that they would use this peak line in the data as a reference to calibrate their actual data. (known as ‘zero-point correction’).

This ensures that all the spectrum lines in the data are aligned and any errors that might have occurred during observation are corrected for.

## Other lines ?

There are other lines at 6300,6363, etc which are sometimes as bright or brighter than the 5577 line that are also used for calibration.

If you are interested in learning more, the following are three papers that this post was inspired from and they dive deeper into more technical details that underlie this fascinating topic:

Have a great day!

# Length contraction , Time dilation and Lorentz Transformation: Pokemon edition

“Most of Special Relatvity is like playing a game of – He said, She said.”

Inspired by the recent Pokemon movie trailer, this post takes you on a journey with Ash and Pikachu  as they find out about the founding principles of special relativity – Length contraction, Time dilation  and Lorentz Transformation.

## Time dilation

Let’s consider a scenario where Pikachu is on a  car moving with a velocity $v$ and Ash is standing the ground.

There is a Pokeball on the floor of the car. It is constantly emitting light straight toward the ceiling, Let us try to understand how Ash and Pikachu would see this photon of light from the time it leaves the floor to the ceiling and back, in their respective frame of references:

How pikachu describes the event:

Pikachu only sees the photon bouncing up and down and therefore:

$2L = c t^{'}$

$4L^2 = c^2 t^{'2} \rightarrow (1)$

How Ash describes the event:

But when the trajectory of the light is viewed from Ash’s frame of reference it is not a straight line at all and therefore, according to Ash:

$2 \sqrt{L^{2} + \frac{v^{2}t^{2}}{4}} = c t$

$4 L^{2} + v^{2}t^{' 2} = c^{2} t^{2}$

$4 L^{2} = (c^{2} - v^2) t^{2} \rightarrow (2)$

Since both these descriptions of what happened are true and have to agree, we must therefore from (1) and (2) have that,

$t = \gamma t^{'}$ where $\gamma = \frac{1}{\sqrt{1- \frac{v^2}{c^2}}}$

This ensures that both the event descriptions are satisfied. The implication being that Pikachu who is on a cart moving with some velocity has to be experiencing time slower than someone who is not moving at all (Ash).

## Length Contraction

Consider the same scenario as before except that this time the Pokeball is emitting light along the direction of motion and back.

How Pikachu describes the event:

If the time measured by Pikachu is slower than Ash from the previous analysis, then we expect that the distance traveled by the photon in Pikachu’s frame of reference will not be $L$. Instead let’s call that distance $L^{'}$. Therefore,

$2L^{'} = c t^{'}$

$\frac{2L^{'}}{c} = t^{'} \rightarrow (3)$

How Ash describes the event:

According to Ash, the photon was initially moving along the direction of motion and then started moving against the direction of motion and therefore the total time taken would be the addition of the time taken in both the cases i.e

$\frac{L}{c+v} + \frac{L}{c-v} = t$

$\frac{2L}{1 - \frac{v^{2}}{c^{2}}} = c t$

$\frac{2L}{c} = \left( 1 - \frac{v^{2}}{c^{2}} \right) t$

$\frac{2L}{c} = \frac{1}{\gamma^2} t$

But, we found from our previous analysis that $t = \gamma t^{'}$ and

$\frac{2L^{'}}{c} = t^{'}$. Plugging these back into the previous equation gives us:

$L = \frac{L^{'}}{\gamma}$

The length of the cart as observed by Ash ($L$) is shorter than that observed by Pikachu ($L^{'}$). This implies that if Pikachu is holding a 1-meter stick,  then Ash would see the 1-meter stick to be shorter than 1-meter. And in general any object that is held parallel to the direction of motion would appear ‘squished’

## Lorentz Transformation

Now consider the final scenario where at $t=0$, the pokeball is at a distance $x$ from both of them and emits a photon of light. Pikachu decides to move towards the pokeball with a velocity $v$ to catch it before Ash does.

This pulse of light would reach Pikachu when he is at a distance $x^{'}$ from the Pokeball and would reach Ash at a distance $x$ since he is not moving.

Length contraction according to Ash:

Having  understood the concept of Length contraction, Ash would say:

$x^{'} = \gamma \left( x - vt \right) \ \ \rightarrow (4)$

Length contraction according to Pikachu:

But Pikachu also understands Length Contraction and thinks that he is at rest and it is Ash who is receding backwards. And therefore:

$x = \gamma \left( x^{'} + vt^{'} \right)$

We must understand that both are valid representations of  what is happening and must agree. Therefore plugging in (4) into the above equation and performing some simple algebraic manipulations we get,

$x = \gamma \left( \gamma \left( x - vt \right) + vt^{'} \right)$

$\frac{x}{\gamma^{2}} = \left( x - vt \right) + \frac{vt^{'}}{\gamma}$

$vt - \frac{v t^{'}}{\gamma} = x \left( 1 - \frac{1}{\gamma^2} \right)$

$vt - \frac{v t^{'}}{\gamma} = \frac{v^{2} x}{c^{2}}$

$t - \frac{t^{'}}{\gamma} = \frac{v x}{c^{2}}$

$t^{'} = \gamma \left( t - \frac{v x}{c^{2}} \right) \ \ \rightarrow (5)$

(4) and (5) are known as Lorentz Transform equations  and relate the distances and time measured in Ash’s reference frame with that measured in Pikachu’s frame.

$x^{'} = \gamma \left( x - vt \right)$

$t^{'} = \gamma \left( t - \frac{v x}{c^{2}} \right)$

One can start off in Ash’s frame of reference and perform a “Lorentz Boost” to see how things are happening in Pikachu’s frame of reference. Often it is convenient to write them in a matrix form:

$\begin{bmatrix} ct^{'} \\ x^{'} \end{bmatrix} = \begin{bmatrix} \gamma & -\frac{v}{c} \gamma \\ - \frac{v}{c} \gamma & \gamma \end{bmatrix} \begin{bmatrix} ct \\ x \end{bmatrix}$

In this post we have constrained Pikachu to move only in the x-direction but this need not be the case. A fun little exercise would be to extend this analysis to Lorentz boosts in 3-dimensions.

It is often more enlightening to see these relations when applied to problems and my personal recommendation would be the book – ‘Spacetime physics by Edwin F. Taylor’. Thank you for joining Ash and Pikachu on this journey. Hope you learned something new. Have fun!

I have always been fascinated by Pokemon.

Tiding through the waves of time, now that I think about it : Pokemon did teach me a lot about physics, especially electricity.

## What is Electricity ?

Electricity stems from a potential difference between two areas,
which allows for electromotive force to ensue in mobile electrons.

## Bio-electricity

In biological cells, a voltage imbalance or a cell potential difference exists between the inside and outside of a a cell.

The cell achieves this by removing 3 sodium ions for every 2 potassium ions allows into the system. The removing process consumes energy ( ATP ).

The sodium ions leaving the cell

The Potassium ions entering the cell

Source Video

## Pikachu and Bioelectrogenesis

Where does pikachu gets it’s electrical powers ?

Its by a process known as bioelectrogenesis.

Bioelectrogenesis is the generation of electricity by living organisms

How it works is rather blunt. Remember I told you that the cells are maintained in a potential difference.

There are passageways /electrolytes that are present that allows a flow of ions through them.

When required,
the brain of the pikachu sends a signal through the nervous system to these
electrolytes, opening ions channels and reversing charge polarity,
causing an abrupt difference in electric potential.

The final effect is the generation of electric current, capable of going up to 100,000 Volts during its thunderbolt move.

Result : Opponent stupefied.

## Water is better conductor than air

Most of animals that bioelectrogenic in nature are aquatic creatures ( electric eels, rays, cattlefish, etc ) . This is because water is a much better electrical conductor than air, therefore electrical signals signals can be transmitted through water.

This betters the chance for the organism to protect itself against predators. Pikachu is not aquatic because probably the writes didn’t want it be so – Poetic License ;P

More:

Some other pokemons that were also bioelectrogenic  were:
Eelektrik and the Eelektross

The voice of Pikachu – must watch

Electrogenic Humans

The one that ash has is a male pikachu. There is a female to the species as well. ( Look at the tail )

That’s pokemon physics for you folks.

Hope you enjoyed reading this post as much i did drafting it. Oh boy! There is physics just in about everything !

# Parallax method, 61-Cygni and the Hipporcas mission

It is trivial for most astronomy textbooks to illustrate the parallax method as follows:

This is absolutely fascinating, but it was really hard to find actual images of stars in books that illustrate this.

This is the proper motion of 61-Cygni, a binary star system over a span of couple of years.

But Bessel discovered that in addition to this proper motion, 61-Cygni also wobbled a little bit from side to side because of the parallax during observation.

The following is a plot of the motion of 61 Cygni – A which beautifully  elucidates the proper motion and the effect of parallax (i.e the wiggle of the blue line with respect to the mean free path)

In addition, if you would like to actually play around with data for yourself, the The Hipparcos Space Astrometry Mission might interest you a lot. The mission was Launched in August 1989 and successfully observed the celestial sphere for 3.5 years before operations ceased in March 1993 employ

The documentation and the catalogue are fairly clear ,  instructive and easy to use. Have fun!

## The Chandler Wobble

Earth precesses around its axis every ~26000 years.

But in addition to this precession, there is an extra wobble that was observed by Kustner and later followed up by Seth Carlo Chandler, Jr called the Chandler Wobble that occurs at a much smaller time scale.

In 1888, Kustner found that the latitude of Berlin had changed slightly
during his observations of the night sky.

Therefore in 1891,Chandler. decided to conduct a 14 month study examining this change. The following is a plot of the spiral path taken by the earth’s axis over that 14 month  period.

The following plot shows the motion from 1909 – 2001.

Although many theories indicate that this is due to the fact that earth is not a perfect spherical rigid body, it is still not entirely clear on the mechanism that drives earth into this small wobbly motion.

If you took a closer look at the plots you would find that this wobble is of the order of a couple of meters which most certainly does not seem like a lot.

But if you are an astronomer if you do not account for this correction, you might just end up pointing your telescope at the wrong object

Have a great day!

## Pillars of creation, Lick Observatory, 2018 [RAW]

Having recently attended a workshop at the Lick Observatory and the opportunity to observe at the telescopes there, this is the raw data of the pillars of creation that we were able to capture using the Nickel Telescope whilst there.

Exposure time: 300 seconds

Location : M16 (Eagle Nebula)

This image needs to reduced even further to correct for the anomalies in
color that one can observe on the image and that’s something we are
currently working on.  We hope to share the entire data with you in a month’s time after post-processing.

Have a good one!