HOW FAR AWAY ARE STARS?

"...Looking at the stars always makes me dream , as simply as i dream over the black dots representing towns and villages on map. Why, i asked myself shouldn't the shiny dots of sky be as accessible as black dots on the map of France"

-Vincent Van Gogh

Study of heavens is older than study of navigation, agriculture and even language itself. It is one of the oldest sciences ever studied . From the dawn of civilisation , humans have wondered about twinkling dots above their heads , far far away from the reach. Study of these twinkling dots has laid down the most important building blocks in arousing the curiosity of mankind into knowing - who we are ?

Stars are all born in nebuli , the cloud of dust. With the gravity working is magic , dust and gas start to collapse. As the temperature rises further stellar evolution continues ; thermonuclear fusion begins, forming helium and we get heat and light .

How far away are stars?

It seems impossible to understand stars, unless one has an understanding of tackling distances between them. Taking the understandings of Greeks in measuring the distance of moon forward , we reached the stars . 

Let's first understand the term parallax - Its the apparent shift in position of an object when it is viewed along different positions. Eg.  hold a pencil in your  hand and extend your arm. Notice the shift in position of pencil as you view it through your one eye keeping the other closed at a time. Your eyes have a certain distance between them that causes this parallax . Similar shift in the position of star occurs when it is viewed from different locations of earth and the astronomers didn't spare this and made it one of the most powerful tools of astronomy !

Let me show you how -

As seasons change, the location of stars in the sky changes - this apparent change in the position of the star when viewed from different points on earth is called - Parallax. 

We take the benefit of this observation and by measuring the position of the star in the sky 6 months apart , by simple mathematics , we get the distance to stars .

we already knew the distance from sun to earth( 1 AU ) 

we measured the angle' alpha 'from earth and putting all in formula - 

alpha= (1 AU)/d  , we get the distance to the star.

But the process is not as straight forward as it looks , twinkling of stars - though soothing to eyes are a major trouble for astronomers because it limits the accuracy to which the position of star is known .

One way to solve this problem is to have thousand of observations to get the accurate position of star and the other is to position our telescopes outside the atmosphere of earth to avoid turbulence due to atmosphere and get the measurement of accurate position. That's why we have most of the telescopes placed higher up in atmosphere.

Everytime we need to measure the position of a star , it's tedious to take thousands of observations . This parallax method surely has limitations . Is there a better way?( yes obviously , else why would i be having this question here ) . It lies in measuring the brightness of star.

How bright are stars and how to measure the brightness?

Let's first talk about brightness in daily life. You go to market and purchase 1000 W bulb. This bulb gives off 1000 Joules of energy in one second. Physicists call this term luminosity.

The other term is apparent brightness. It describes how a star gets dimmer and dimmer as it gets far away. We use this relationship between distance and brightness to a great extent in astronomy.

 How Astronomers measure the brightness?

Stars radiate enormous amount of energy so thy don't use units like Watts cuz that will be very inconvenient. 

The apparent luminosity of of a star is descirbed as

 apparent magnitude (m) and

                                                 

 total luminosity is described as absolute magnitude (M)-magnitude of an object placed 10 pc     away.

                                                 

 Where parsec(pc) is the distance which a star has to travel for it's parallax to be 3.26 light year(light year is equal to the distance light travels in one year = speed of light(m/s) * no. of seconds in an year)

The construction of magnitude scales was done on the basis of the most fundamental logic-

Our eyes are more sensitive to geometric than arithmetic progression.

This fact can be understood very easily as you will be able to distinguish more clearly in between two bulbs if brightness of one bulb is greater than that of other by a factor 5 i.e b1=5*b2

 than if the brightness of one bulb is 5 + greater than the other , i.e b1=b2+5.

There are two things which we need to note about these magnitudes here -

The scale runs on logarithmic scale ( magnitude of one star is dimmer than other by same factor)and backwards( higher the magnitude , dimmer the star )

Here is the mathematical expression for apparent magnitude- 

m1 - m2 = -2.5 * log( B1/B2) where m1 , m2  and B1 , B2 are magnitudes and brightness of objects respectively. 

and here is mathematical expression for absolute magnitude- m-M = 2.5 log (d/10)^2

 If the object is at a distance d pc, then (10/d)^2 is the ratio of its apparent brightness and the brightness it would have if it were at a distance of 10pc. 

Starlight deciphering the mysteries of cosmos

Stars send us message encoded in the starlight. We decode this message by studying the spectrum of stars, we can encode many things, like what is the star made up of , the age of star .

 

Suppose two stars have same spectrum, one is near and one is very far away . If spectra of two stars is identical then experience has shown astronomers that the two stars will also be very similar in their other properties, such as their mass, radius, and total luminosity. If it turns out that the second star has apparent luminosity that is only one ninth that of  1st one , then it is likely that this is because the star is thrice as far away, so that its light is spread over a sphere whose area is nine times as large as that of the first one. Noticed how cleverly we figured out distance here ? This is what astronomers use to take account of very large distances.  

well..this is not the end to the story still , we got a lot of information from a single twinkling speck on the sky , that realisation is awesome in itself. I wonder , what it would be like decoding every single bit of information that is given to us ! Our ancestors were definitely excellent at it ; we owe our understanding to all those curious , passionate and brave minds who decoded the biggest mysteries without any basic tools. There is still a long long way to go .. Every piece of the sky tells a story , just be curious enough to decipher one.

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INTERPLANETARY TRAVEL

"The first day or so we all pointed to our countries . The third or fourth day ,we were pointing to other countries . By the fifth day , we were only aware of earth"-Sultan bin Salman Al-Saud

Curiousity about space is the essence of human existence. So , this blog post is about some of the tricks that we are using for interplanetary travel.

Well why do we need to use them though ?

Because they are more efficient . Planets, let alone galaxies are millions of kilometres away from us and exploring them requires efficiency in fuel and time . This is what is provided by these tricks.


Getting away from Earth ; The pale blue dot

Want to escape from solar system , make the total energy of trajectory = 0

What's total energy ? It's the sum of -

Kinetic Energy - energy due to motion +
Potential Energy - energy due to configuration
Putting total energy 0 we get escape speed from Earth = ( 2* G *M /R) ^ (0.5) where ,
G = Gravitational Constant -6.6× 10^ (-11) Nm^2 kg^2
M = Mass of Sun = 1.98 * 10^30 kg 
R = Distance of sun from Earth = 152.09 million km

Putting all these values we get escape speed from earth - 11.2 km /s 

Long story short, The spacecraft launched in any direction will reach the out of the Gravitational pull of earth when launched with the speed 11.2 km/s.


Getting to other planets :

Hohmann Transfer

Credit:Wikipedia

Hohmann ( a German scientist) showed that the lowest propellant path between any two orbits is an elliptical orbit which forms a tangent to the starting and destination orbits.
(Pic)

Insight - Spacecraft to Mars and Chandrayaan-Spacecraft to moon used this technique

Calculating launch speed to other planets

Suppose we launch from the Earth, at a distance R1 from the Sun, at a target planet a distance R2 from the Sun. 

Let r denote the ratio R2/R1; then r is just the orbital radius of the target planet expressed in astronomical units (AU).
v⊕ = orbital speed of the planet from where the spacecraft is to be launched 

To reach the outer planet, the spacecraft is required to have a speed relative to the Sun after escaping the Earth of:
v = (2r/r + 1)^ 1/2 * v⊕ 

For Jupiter we have this speed = 38.6 km /s
To get launch speed from Earth , subtract earth's orbital speed 29.8 km/s and add the escape speed.
Thus launch speed to Jupiter is 20 km/s 

Similarly, launch speed to a planet from any other planet can also be calculated.


Gravitational Slingshot -

 Using gravity of planet to reach to other planets

Now we have successfully sent our spacecraft to Jupiter , we will gravity of Jupiter to send it further and out of Solar System.
Wait ! If we send a spacecraft on a trajectory
around Jupiter , and we follow its orbit as it falls towards the planet and then comes
back out, we always find that it returns to the same place as we started it with exactly
the same speed. There is no gain! But we see Voyagers speeding up gaining speed from Jupiter. How's that possible?
Well answer lies in the 'frame of reference' .
We need to look at the change in speed through frame of reference of Sun:

It seems like breaking the conservation law, it seems like planet is taking energy out of the system out of nowhere . Well that's not actually the case .

The gain in speed of spacecraft is accompanied by slowing down of Jupiter a tiny bit , 

because Jupiter is very large , this slowing down isn't prominent enough . But yeah , with millions and trillions of similar slingshots , we can slow the planet to even such an extent that it will start dropping towards Sun's gravity .

This is pretty much what Voyagers have used to escape the boundaries of solar system.
Taking spacecraft approaching along different direction , we can slow down the spacecraft as well and make it fall close towards sun and speed up the planet.

Thus we see how we can use planets like controllers for our journey to the outer world.This realisation so freaking awesome in itself !
Not the gravity but time dependency of gravitational field due to motion of the planet has helped us in taking some advantage out of it . The conservation laws you see , are built on principle of invariance , invariance of gravitational fields , time .

The relation between conservation laws and invariance is built into the laws of physics at their deepest level. How can we mold them into our benefit is totally upon our understanding of phenomenon and curiosity in understanding the nature of Universe .

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For astrophysics geeks, i would here like to mention the list of top 25 astrophysics blogs of 2020 by feedspot. Go check them out by clicking here- top25astrophysicsblogs

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