kepler's laws

The front side of my equation sheet for the last mechanics exam. (Scored 20 points over the average/median on that one, aw yis.)

24 days of Books (1)

There are 24 days to Christmas and I thought I would do something a little bit different. So, everyday till Christmas I will post a book recommendation. It will be a math or math related book of course ^_^ Enjoy

First is Mathematical Principles of Natural Philosophy

Philosophiæ Naturalis Principia Mathematica (Latin for “Mathematical Principles of Natural Philosophy”), often referred to as simply the Principia, is a work in three books by Sir Isaac Newton, in Latin, first published 5 July 1687. The Principia states Newton’s laws of motion, forming the foundation of classical mechanics, also Newton’s law of universal gravitation, and a derivation of Kepler’s laws of planetary motion (which Kepler first obtained empirically). The Principia is “justly regarded as one of the most important works in the history of science”.

The French mathematical physicist Alexis Clairaut assessed it in 1747: “The famous book of mathematical Principles of natural Philosophy marked the epoch of a great revolution in physics. The method followed by its illustrious author Sir Newton … spread the light of mathematics on a science which up to then had remained in the darkness of conjectures and hypotheses.”

Level of mathematics: I would say a high level, this is basically calculus. 

Astronomy/Astrophysics Study Tips

So I noticed that Astronomy was one of the subjects in high demand on the @studyblrsubjects blog, I’ve decided to write a few study tips. One of my majors is astrophysics and I actually do pretty well in it (Alhamdulillah), so hopefully these tips will be good.

The set-up of astro courses at different universities vary quite a lot. Some are more practical, some deal a lot more with the theoretical and hard physics side, others are more balanced. If you’re interested in studying astronomy or astrophysics, it’s a good idea to contact the relevant department at your university of choice and find out what their programme is like.

1) Be careful with conversions. Astronomy uses its own units. Parsecs, Astronomical Units, solar masses, magnitudes, e.t.c. are defined in terms of what’s practically measurable when making observations. However, things like Kepler’s Law and Newtonian gravity and most formulae are defined in terms of your usual SI units. Learn how to convert correctly and check which units are the correct ones to use. It’s really useful to include the units when you’re substituting values into an equation so that you can use dimensional analysis to see if you’re missing a conversion. And don’t worry - everyone messes up a simple magnitude calculation every now and then I can rant about the awful magnitude system for ages.

2) Revise your trigonometry. Remember the Sine Rule and the Cosine Rule from high school? Me neither. Brush up on your trig because it is used all the time in astronomy. The last thing you want to do is incorrectly define a parsec because you can’t remember which sides correspond to sin and which correspond to tan. Be especially careful when converting between radians, degrees and hours. 

3) Use a compass. This is non-essential, but it’s useful to have one because you quite often need to draw a LOT of circles.

4) You don’t have to know constellations to study astrophysics. I can spot the Southern Cross (a.k.a. Crux) and Orion in the night sky. That’s about it, and I get As for astrophysics and astronomy. It’s pretty cool if you are able to spot constellations, and if you know where to look for interesting nebulae and star clusters, but it’s not essential. Also, if you want to bullshit your knowledge of constellations other people will believe you simply because you’re an astrophysics major.

5) When in doubt, the answer is probably spectroscopy.

6) Ask additional questions. Space is pretty cool. Whenever I get bored in class, I pay a little extra attention and ask questions on things that are vague. It gives you more insight into the unanswered questions in astrophysics (there are a LOT of unanswered questions) and into how astronomical research is conducted. Or you’ll learn some fun facts about space.

7) Set your wallpaper as a Hubble Space Telescope picture. Again, non-essential, but why wouldn’t you want to?

8) Like with any type of physics, diagrams are always useful.

9) Make sure that you can derive things on your own. Sure, some of the formulae you’ll be working with are based off empirical evidence or are beyond what you’re expected to know. However, there are plenty that come from the basics you learnt in physics and you should be able to derive those. 

10) Use the internet. Crash Course Astronomy (or even the show Cosmos) can help you familiarise yourself with the terminology and processes in astrophysics. Knowing the “story” behind processes will make it easier for you to concentrate on the actual physics behind the processes. For example, if you know that a black hole can form from the collapse of a dying star from listening to Dr Neil Degrasse Tyson explain it, you can concentrate on deriving the schwarzschild radius instead of trying to wrap your head around the idea of a black hole. Or, if you know that famous Carl Sagan quote about us all being made of starstuff, it’s easier for you to concentrate on understanding the different stages of star formation that leads to the creation of carbon.

I hope these tips are helpful. If you have any questions, feel free to send me a message or an ask! Happy studying.

xx Munira

I like to print out PDFs of all my textbooks, comb bind them, and then go on a walk and study them at the same time and bleed all over them with gel pens.

Integration, Relativity, and Kepler's Law

A/N: Drabble I couldn’t pass up on for imaginexhobbit’s “Imagine telling the Company about technology advancement and complex mathematics.” I went a little into science, too, but…I couldn’t help it. Sort of ThorinxOC if you want to look at it that way. And I didn’t name the character. Cause I suck at names.

Oh, and also on AO3.

    Despite it all, Middle Earth and the Journey undertaken by the Company of Thorin Oakenshield did not turn out to be everything which she had expected. As much as she had brushed it off before, her entire generation was so damned dependent on modern day commodities. What was the saying? You never know how much you loved something until you lost it? Something like that.

    "You alright, lass?“

    She looked up from where she lay against the trunk of a thick tree, stilling her fingers tapping against the hard cover of her textbook. "Yeah, I’m all good, Bofur, thanks. Just miss home a little.” She thought an extra second, only then realizing who she was speaking to. Honestly, she had nothing on the dwarves; they hadn’t seen their home for a far longer time than she, so there really wasn’t any right for her to even try to complain. “I mean, nothing like you guys, nowhere even close. Just, running water sounds nice right now.”

    The dwarf quirked his head a bit, his mustache swaying with the movement. “There’s a river right around here if you need it. I’d be happy to show you -”

    “Oh!” she laughed slightly at the dwarf’s confusion. Although, she had to admit, she’d taken a liking to Bofur; he was a nice enough character, and Tolkien didn’t do him near enough justice.” No, I don’t mean like that. I mean, back home, we’ve got pipes and stuff in our homes to bring water to us to bathe and everything like that. So much easier. And warmer too!” She leaned her head back, thinking of her suburban house and family, before turning to find the confusion on her companion’s face. “But you probably don’t want to hear about all that. Me and my pathetic little problems.”

    “No, nothing like that, lass!” Bofur broke out of his trance, crouching down next to her now. “Just puzzled, is all. Is your entire world like this?”

    “Well…” The smile spread across her face as she contemplated where to begin. “Basically, yes.”

Keep reading

I would like to File a Formal Complaint with Wolf 359.

I am a physics PhD student, who has very much enjoyed this narrative program. However it did at first cause me some difficulty in my work as it becomes very hard to focus on the lecture when the phrase “Hilbert Transform in Minkowski space” is said unironically by the professor when you aren’t expecting it in the middle of a derivation. But, I got used to it, partially because I just don’t interact with that branch of things very much. It was all fine and dandy until…

Maxwell.

MAXWELL.

Dr. Alana MAXWELL

Jacobi Matrices I could have avoided. Even Kepler’s Law is such a narrow case that I could laugh and move on but you had to taint MAXWELL’S EQUATIONS? THE FUNDAMENTAL LAWS OF ELECTROMAGNETISM??? I LITERALLY USE THEM EVERY DAY!!!!!

Will you Monsters Leave me no peace???

This image displays Kepler’s second law of planetary motion.

“A line joining a planet and the Sun sweeps out equal areas during equal intervals of time” (Meaning that each triangle seen there has equal area.)

  • The black dot represents a planet, the point where the black lines intersect represent the sun.
  • The green arrow represents the planet’s velocity,
  • The purple arrows represents the force on the planet.

Mars-Sized Exoplanet Discovered

An interesting method has revealed the mass of a small planet orbiting a faraway red dwarf star: Kepler-138.

The planet in question, Kepler-138b, had a known diameter and volume but getting mass is no simple task, especially for something that causes so little a tug on its host star.

Astronomers therefore decided to look at how the planet tugged on other planets.

Planetary orbits are extremely predictable. It’s how NASA can send a spacecraft all the way to Pluto, launching it on the correct trajectory almost a decade before it will even arrive.

It was discovered that two of the other planets around Kepler-138 had orbits that were somehow off by an hour.

Bingo. By observing how much their orbits shifted SETI Institute astronomers were able to decipher the force of gravity the small planet was exerting upon its neighbors.

The force of gravity can then be put in these terms: 

Fg=G(M1M2)/r2

With the important parts being r, G and the two masses. G is just a constant. r is known thanks to Kepler’s third law of planetary motion. The mass of each of the two larger planets was known already (larger exoplanets are easier to observe and work with).

With this equation, force of gravity becomes dependent on two things: distance and mass.

As any grade school student can tell you, if you have an equation and you need to solve for x, you just move x to one side (or in this case one of the “M”s). This is all they did here and it resulted in a groundbreaking, beautiful discovery.

What we now know is that this planet is rocky and likely shares some fundamental similarities to Mars.

(Image credit: Danielle Futselaar)