Astronomy

                       Mathematics is the language in which God has written the Universe 

- Galileo Galilei 

Overview

Astronomy is my favorite hobby, and nothing brings me more joy than understanding or explaining astronomical phenomena through mathematical analysis. Whether observing the night sky with a telescope or binoculars, I view everything through a mathematical lens. This page contains my astronomy-related publications, notes, talks, calculations, observations, experiments, and astrophotography images. These items showcase how mathematics can help us understand the cosmos.

As a member of the Twin City Amateur Astronomers (TCAA), I've gained a wealth of knowledge in astronomy. I've served as a board member and president of this great club, which has been a valuable resource for me. Additionally, some of my research and outreach activities in mathematics and astronomy have been generously supported by the mathematics department at Illinois State University (ISU). I'm grateful to TCAA and ISU for their support.

Publications and Notes

These are some articles I wrote on Math and Astronomy. High school students can profitably read them.   

My Astronomy Blog

I started recording my observations and sharing fun facts and discoveries on my astronomy blog.  Here you will also find my monthly notes I wrote as a president for our astronomy club's newsletter "Observer."

Talks on Astronomy 

Here are some recordings and PowerPoint slides of talks I gave on Astronomy (from a mathematical standpoint) for the TCAA club, ISU Physics Colloquium, and ISU Undergraduate Math Colloquium.

How do we determine directions, latitude, and longitude?

How do we measure distances in the universe?

Math and Astronomy - I

Math and Astronomy - II

Mathematics and Astronomy.ppsx

ISU Math colloquium, Nov. 2017

Measuring distances: from ancient Greeks to Newton

distances-universe-TCAA.ppsx

TCAA public talk, April 2019

Measuring distances: from ancient Greek to modern astronomy

Time Reckoning_ Astronomical Clocks and Calendars.ppsx

TCAA Meeting, March 18, 2021

Time Reckoning: Astronomical Calendars and Clocks

Abstract Telescopes.pptx

        ISU Astronomy Club, April 28, 2022

Abstract Telescopes

Copernican Revolution

TCAA, Public talk, April 22, 2023

Copernican Revolution

My Favorite Books on Astronomy

Here are  my top 10 in no particular order.

Astronomy Clubs

I benefit from the membership with the following astronomy clubs/planetarium. You can find them on Facebook as well.

Awards

NASA Night Sky Network Award


Eugene & Donna Miller Family Award

The Eugene & Donna Miller Family award of TCAA  recognizes the exceptional efforts made by one or more parents to inspire and cultivate their children's passion for astronomy.

I am grateful to be a recipient of this award in 2023 for the work I have done with my daughter Shloka.


My Astronomy Equipment

Celestron 130mm SLT Newtonian Telescope

Most of my ventures until 2022 used this scope. I bought this in 2017, and I have been having great fun with it. I use it mostly in my backyard but also take it out to some relatively dark sites for star parties.

I also took some pictures (mostly the Moon) with this basic equipment. See below. Some of these amateur images made their way to the front covers of journals and astronomy newsletters.

This is a perfect beginner go-to reflecting telescope. Easy to move around, and you can set it up in 5 minutes. Highly recommend this to those who would like to get their feet wet in astronomy.

 CANON  12 x 36 IS Binoculars

This is a recent addition to my gear (2019). It is an excellent pair of binoculars for the armchair astronomer. My best stargazing experience is to lie down in my backyard hammock and enjoy the night sky with these binoculars. The IS (image stabilization) in these binoculars is the most extraordinary feature!

 Celestron CPC 11-inch Telescope

After spending more than five years with an  SLT 130 mm scope, I am excited to work with this new Celestron CPC 11-inch telescope. 

This picture was taken at TCAA's public viewing session at the Sugar Grove Nature Center (SGNC) where  I was interviewed by Pentagraph - our local newspaper - to feature our club and our activities in news: 



Messier Programs and Astronomy Observations

Spring Mini Messier Marathon (May 12th, 2021)

Summer Mini Messier Marathon (Aug 16th, 2021)

Fall Mini Messier Marathon (Oct 27th, 202)

Binocular Messier Marathon (in progress)

 Astrophotography 

Almost all images below were taken from my backyard using my Celestron SLT 130mm and Google pixel phone in astrophotography mode. 

Mare Crisium

September 2, 2019. Sea of Crises: about 345 miles in diameter.

1. The last spacecraft of the USSR to make a soft landing (Luna 24) on the Moon was in this region (August 18, 1976).

2. There are three prominent craters in the interior (zoom in to see these) that are named after (in order from top to bottom):

Jean Picard (French astronomer),

Benjamin Peirce (American Mathematician and Astronomer)

Lewis Swift (American Astronomer)

Hunter's Moon

October 13, 2019. It gets this name from hunters back in the day who killed their prey by autumn moonlight and stockpiled food for the winter ahead. It is the full Moon after the harvest moon.

The full Moon was too bright; even with a 13% filter, I could not bring the detail I wanted.

Harvest Moon

Sept 2019. My photograph of the Harvest moon was chosen as the "Image of the Month" for the front page of the October newsletter of our local astronomy club. Also, in this issue, under Image gallery, they published some other photos I took recently, including lunar craters, Orion Nebula, and my incredible Celestron Telescope in my backyard.

Lunar Eclipse

January 2019. The picture was taken when the Moon was mostly in the umbra; note the gradual transition in color from the reddish umbral part to the bluish penumbral part. This picture was featured on the cover page of Math Horizons.

Full Moon

August 15, 2019. It is was a cloudy night, but it was just the right amount of cloud to act as a natural moon filter.

Copernicus and Eratosthenes craters! 

October 7, 2019. 73% waxing gibbous Moon. The most prominent crater you see in these pictures along the terminator is Copernicus, and right next to it, to the right side, is Eratosthenes. 

Both are towering figures in astronomy. Copernicus is a polish astronomer who is well known for his revolutionary work on the heliocentric model, and Eratosthenes is an ancient Greek polymath who is well known for his ingenious experiment where he measured the circumference of the Earth to a remarkable degree of accuracy.

         Arundati-Vashistha or Mizar-Alcor

Splitting a double

September 23, 2019 (the night of Autumnal equinox): My first "splitting-a-double" exercise. Even though this is a poor image, the experiment was a lot of fun! 

 

The target was the famous Arundhati-Vashistha (wife-husband) nakshatram or Mizar-Alcor star. This is the 2nd star from the end of the handle of the Big Dipper. These are the first stars that many Indian couples see in a long series of "stars" to follow (pun intended). Below I summarize my findings and some not so well-known facts.

When I looked at this pair through my star-finder for alignment, I only saw one star.

Next, when I looked through my 25mm eyepiece, I saw the pair very clearly: Vashistha (Mizar) and its faint companion Arundhati (Alcor), which are separated by 12 arc minutes.

 But after I replaced 25mm with a 9mm eyepiece, to my surprise, I saw a further splitting of Mizar. Mizar is a binary star! I saw Mizar A and Mizar B.

I read that this was the first binary star that astronomers discovered.

It turns out (this I found out later)

remarkably both Mizar A and Mizar B are also binary stars! So Mizar is a quadruple star!! Alcor is also a binary star. So what we have here is really a sextuple (6 stars) system!

In the picture, the star at the top is Arundhati, and the bright star off to the right is Vashistha.

Arundhati and Vashistha are NOT known to be gravitationally bound. They are some evidence that they may be gravitationally bound but not fully confirmed. In fact, they are separated by three light-years.

Orion

This is one of my favorite pictures. My telescope in the foreground and Orion in the background during wee hours. The Orion nebula is also visible if you take a closer look.

Saturn

August 6, 2019: My first picture of Saturn, using a Barlow lens along with a 9mm eyepiece.

M 44 - Beehive open cluster  

This is in the constellation of Cancer, and it contains about 1000 stars that are gravitationally bound, but my picture only shows about 40 bright stars. Estimated to be 600 million years old, and it is at a distance of 500 light-years from Earth.

Other interesting facts: M44 is one of the first deep sky objects studied by Galileo in 1609. The first planets that were detected (2012) orbiting stars like Earth's Sun were in this cluster.

 The Straight Wall

May 20, 2020. The one attached here was taken with a 5mm eyepiece and a 13% filter. You see the great wall, and lots of craters are clearly visible, especially the ones along the terminator. You really see the scars of time!

Interesting geometric fact: When the Moon is its first quarter phase, the Earth, the Moon, and the Sun form the vertices of an imaginary right triangle with the Moon at the 90-degree vertex!

Sirius

September 30, 2019. Sirius (aka Dog Star) - the brightest star in the night sky! It is a binary star, but I could not resolve it. At a distance of 8.6 light-years, it is scorching with an apparent magnitude of -1.46 in the constellation of Canis Major.

It is twice as bright as the second brightest star (Canopus)and 25 times more luminous than our Sun. It is one of the stars on the flag of Brazil.

Waxing Gibbous

Sep. 10, 2019

Great conjunction of Jupiter  and Saturn

 December 21, 2020: The night sky offered this beautiful astronomical treat for stargazers worldwide. It was an exceptionally rare conjunction of Jupiter and Saturn that brought them to 0.1 degrees of angular separation -- a fifth of the full moon's diameter. It marked their closest approach since 1623 and the closest visible conjunction since 1226 (almost 800 years ago!). 

I took this picture using a Google pixel-3 camera and Celestron 130 mm SLT Newtonian Telescope equipped with a 25 mm eyepiece and a 2x Barlow lens. For the first time, I saw these two planets in the same telescopic field of view.  

Shades of the Partial Lunar eclipse.

I enjoyed the longest partial lunar eclipse in over 500 years. This picture was taken on Nov. 19th, 2021  from my front yard. There was an annoying street light which made it hard to capture a good image. My best attempt despite this obstacle. 

 Astronomy Calculations 

Being a mathematician, I love doing calculations in astronomy. Here are a couple of examples. 

 Galileo's Experiment

May 22, 2018: I had a lot of fun in my backyard looking at the Moon. Here is a view through my Celestron Nexstar 130mm SLT (5mm eyepiece and 13% moon filter).

It was Moon's first quarter, and the viewing conditions around 10:00 PM CST in Bloomington-Normal were perfect for examining the topography of the lunar surface. I saw craters, mountains, and the shadows of these objects on the Moon. The best view of the Moon I have seen in my telescope.

I also had the pleasure to reproduce a famous experiment done by Galileo back in 1609 to estimate the height of some mountains on the Moon.

The idea is simple but ingenious. Rays of the Sun at dawn hit the mountain peaks before they hit the base, hence the gap between the terminator (the imaginary arc dividing the bright and dark sides of the Moon) and the tiny triangular light mark in the picture to the right of the terminator. Moreover, the rays hit the surface of the Moon tangential near the terminator. So a straightforward application of the Pythagorean theorem gives the following formula for H (the height of a mountain)

H = r [sqrt( 1 + 1/ d^2) -1],

where r = true radius of the Moon and d = ratio of the apparent radius of the Moon to the apparent distance between the terminator and the tip of a mountain. (Proof is left as an exercise to the reader.)

d can be computed from the digital image directly. I got d = 15. r = 1080 miles (an estimate well known since antiquity). These values gave me 2.39 miles for H (the height of the mountain)! 

Can you find Vikram using your telescope?

After ISRO's Vikram lander failed to make a soft-landing on the Moon on September 6, 2019, several friends and students have curiously asked me (some kidding and others serious): Can you find Vikram using your telescope?

The short answer is No. You will need a telescope with a VERY LARGE aperture to resolve an object the size of Vikram (which is microscopic on the planetary scale). In fact, the math shows that the aperture of the telescope should be at least 100 meters. To put things in perspective, the famous Hubble space telescope has an aperture of 2.4 meters, and the world's largest telescope (under construction in Chile) has an aperture of about 40 meters.

Here is a long answer for the mathematically inclined (I got all numbers from Wikipedia):

a = Vikram’s length (2.5 m)

A = distance to the Moon (3.6 x 10^8 m)

So the angular separation of Vikram, as seen from Earth, would be approximately

u = arctan(a/A) = 6.9 x 10^(-9) radians.

So the question is, in order to resolve an object with such a small angular separation, how big must be your telescope's aperture (D)? This is given by a formula in optics:

D = L/u, where

L = wavelength of visible light ~ 650 nanometers = 6.5 x 10^(-7) m

u = angular separation in radians.

Plugging the numbers, we get

D = [6.5 x10^(-7)]/[6.9 x 10^(-9)], which is approximately 100 m!!

Most amateur telescopes have apertures in the range of 3 inches – 30 inches, which are far less than 1 m.

Conclusion: No earthbound telescope that has ever been built can resolve the Vikram lander!

Some Experiments

Some fun day-time astronomy experiments using stones and sticks.

Determining directions from the sun

The tip of the shadow of a vertical stick on my driveway moves as the Sun moves across the sky. Recording this position in intervals of 15 minutes by placing stones gives approximately the east-west direction. Perpendicular to that will be the north-south direction.

Measuring latitude from the sun

To get your latitude from the Sun, you need the angle of elevation at local noon and declination for the Sun. Finding the angle of elevation is easy, find the ratio of the length of a vertical stick and take its inverse tangent. Finding declination and latitude involves more work. See my talk on celestial navigation for details.