Two-Body Orbits: Where's The Centre of Mass?

Second Authors: Nathan, Lauren

Introduction

Consider a problem of a planet orbiting a star.  It's easy to see from Newtonian gravitation that they exert a force on one another.  But then, how, according to Kepler's Third Law can we say the planet orbits the star alone?  The star cannot remain fixed while feeling a force.  In order for Newton's Laws to hold, we must say that they both orbit a mutual centre of mass.  Using conservation of momentum, we can determine how far each body truly is from this centre of mass.

Methods

In order to balance forces, we notice that the planet and the star must be at opposite ends of their orbits at all times as seen in the following picture (not to scale).  It follows from this that they have the same orbital period and the same angular velocity.

We know that linear momentum is equal around the centre of mass, such that:



Dividing through, we get the relationship



Rearranging and using the mean semimajor axis, a, we can see that

Conclusion
We have shown here that star does indeed orbit the centre of mass, just as the planet does.  However, looking at these equations carefully, we find that except for very massive planets, the semimajor axis of the star's orbit is roughly zero and the semimajor axis of the planet's orbit is roughly equal to the mean semimajor axis.  As a result, we find that we can in fact use the assumptions implicit in Kepler's Law.

The Death of a Star

Second Authors: Nathan, Lauren


Introduction
We know how stars are formed to a certain extent and that while they are on the main sequence, they are supported by hydrostatic equilibrium. However, when a star moves off the main sequence and can no longer support itself, what happens? We assume that at this point, the core of the sun has converted all of its mass to energy and is now undergoing gravitational collapse.

Methods
We know that the Sun generates energy throughout its lifetime at a rate of:

   

Assuming that the sun uses up the entire mass of its core (10% of a solar mass) as it undergoes fusion, and converts energy with 0.7% efficiency, we can determine the total energy it produces in its lifetime with

 

Dividing this number by the rate of energy production, we can determine the time it takes for the Sun to use all of its mass available for fusion. This time is

 

Now we know the core will collapse, but it won't collapse indefinitely.  We find that the core collapses to the point that the interparticle spacing is on the order of the De Broglie wavelength.  Since it is easy to see that electrons have greater momentum compared to protons of equal energy, electrons are the first to reach this critical density.  We can calculate this using the equations:

It is easy to tell that we have one molecule per cubic lambda.  So we have:

The actual value is 8 times this, for reasons I can't remember, but Nathan tells me Professor Johnson said the factor of 8 was okay to include in our calculations.  So multiplying this by the mass of a hydrogen atom and using T = the temperature of the sun's core, we get density

Which is more than twice the current maximum density of the sun's core.

Conclusions
We find that by converting 0.7% of the mass of the sun's core into energy, the sun's lifetime is roughly 10 billion years, which agrees with what scientists have predicted.  The density of the core after collapse will also be far greater than the current density of the sun's core, which is reasonable or it would not be able to support the sun post-collapse.

An Interview With an Astrophysicist: The Postdoc

As some readers may know, this past summer I had the privilege of being able to work for Dr. Andy Goulding at the Smithsonian Astrophysical Observatory.  Dr. Goulding is a first year Smithsonian Research Fellow in the High Energy Astrophysics division.  He did his PhD work in AGN activity at the University of Durham, UK in the fall of 2010 before moving to Boston as a postdoc.  Working with him was a wonderful educational experience that I'd happily do over if I had the chance.  He gave me a look into the life of a researcher and how much research differs from course work.

We've vaguely kept in touch since September and he generously agreed to answer a few questions on his career for me.  Of course, given the title of this blog, the first question was obvious.  He took a lot of time in answering these questions, and I definitely learned some new things.  For example, I had no idea that there was a difference between a postdoctoral research associate and a fellow.


What is the difference between an astronomer and an astrophysicist at this point in time?  Which, if you have a preference, are you?

From a professional point of view, this is really semantics. People have degrees/PhDs in astronomy and/or astrophysics - it depends on institution. However, it is more likely that someone who is an amateur (non-PhD) is considered to be an astronomer. Classically, an astrophysicist attempts to understand and interpret the astronomical observations through application of physics. My PhD is in astrophysics, so in the strictest sense, I am an astrophysicist.

Star Formation: Timescale and Stability

Introduction
Star formation is governed by the collapse of a cloud of particles into a gravitationally bound sphere which we call a star.  The radius of the could at which this occurs is called the Jeans Length, where the gravitational force of the cloud overcomes the thermal energy causing it to expand.  Here we examine the time scale of such a collapse and also calculate the Jeans Length.

Methods
In order to determine the time it takes for this collapse to occur in terms of the mass and size of the cloud, we consider a cloud of mass M and a test particle a distance away from it.  We assume the cloud has a mass given by

where r is the length of the major axis for an elliptical orbit of eccentricity 1.  By assuming such a geometry for the free fall, we can initially approximate the orbit to a straight line with a  mass M at one end and our test particle at the other.  Since this is a free fall, we can also approximate the time t_ff to be half the orbital period we get from Kepler's 3rd law (a = 1/2 r)

 

Substituting our mass formula into this equation, we get

The implicit assumptions are that we can even call this half an orbit, as an eccentricity 1 orbit is parabolic and therefore not periodic, and that we can approximate this orbit to a straight line.  Now in order to find the Jeans Length, we equate this to the dynamical time, or the time it takes a sound wave to cross this distance.  Let's define this as

Equating the two, we get the radius at which the cloud will undergo gravitational collapse

For an isothermal gas of constant density, this length signifies the minimum radius at which it will continue to be a gas and not collapse into a much denser formation.  This is the Jeans Length to an order of magnitude.  The actual formula for the Jeans Length is

Conclusions
We have hear calculated the free fall time for star formation as well as the radius at which the gravitational force between interstellar dust particles takes over.  It is important to note that since the density is radius dependent, the Jeans Length is not constant for all star forming clouds, but varies even with the change of radius due to collapse and we have

If we consider a cloud that starts out at the Jeans Length for its particular conditions, by the time it reaches half this radius the Jeans Length has decreased by a factor of √8.  As a result, the initial Jeans Length may actually govern how far the cloud will collapse for a given mass and radius.

Becoming An Astronomer

We were recently, or not so recently—I'm very good at procrastinating—assigned a multi-part blogging to find out what it truly means to be an astronomer.  I realise as a sophomore astrophysics major that I still don't understand the specifics of what being an astronomer entails or means to me.  To quote my friend Alexa,

I just want to be one. So much. “Space” is, if you think about it, everything but Earth. When we study it, we’re pausing our narcissistic tendencies for just a moment. We’re not everything; we’re part of everything. Ignoring that is shameful.

She stated in the best way possible what attracts me to astronomy, but that still doesn't mean I know what astronomy is.  Right now I just think of astronomy as some nebulous loosely defined field of Things I Would Like To Do Because They Are Amazing, but that's not an acceptable answer to the question.  So without further delay I shall attempt to synthesise my thoughts on the topic.

Is There Life On Maaaars?

I've been having an uncharacteristic moment of curiosity lately, and that curiosity is about life outside Earth.  Usually I don't care.  I'm much more of a "let's explore and discover the physical laws of the universe" kind of guy.  But today, it's all about life out there, and why not?  Some pretty interesting things have happened in the last week.

1. ESA's Mars500 Simulation Ended
So I have to admit, I knew nothing about this project until I read the article today.  Doesn't prevent me from thinking it's amazing.  In short, a crew of 6 was stuck together in an in-lab "spacecraft" for 17 months, performing the tasks necessary for a real mission to Mars including "entering" orbit and "landing" on Mars.  Conditions were controlled exactly as if they were actually travelling and they completed experiments on the problems brought about by long space missions.  Maybe this will open up opportunities for an actual space mission to Mars after studying the physiological and psychological effects of longterm isolation.  Very cool.  Here is a compiled video diary of their time during the simulation:

2. A New Way to Look for Aliens

Avi Loeb and Edwin Turner of the Harvard-Smithsonian Center for Astrophysics and Princeton University, respectively have suggested a new way to look for extraterrestrial intelligence: doing it the same way we find civilisation on earth.  They intend to look for the lights from their cities.  These two operate on the assumption that life evolves in the light of the nearest star and that any intelligent life forms would have learned to make light and extend their days.  They would have to find a way to filter out the light from the star.  They suggest that one method of doing this is to look for bright areas in a dark phase of the planet's orbit (think of the dark side of the moon).  Unfortunately, this method would require far more powerful telescopes than we now have, but it's definitely a start.

3. Organic Molecule "Sweet Spots"

This isn't technically, astrophysics, however I think it still has a place in a post about life outside Earth.  Astrobiologists at Rensselaer (one of the reasons I didn't apply there was I couldn't spell it on the first try) have discovered areas of higher methanol concentration surrounding some, but not all, newly formed stars.  Methanol is apparently one of the precursors to more complex organic molecules which may give rise to life.  They call this a "sweet spot" of physical conditions that allow these organic molecules to form.  Even more interestingly, from studying concentrations in comets, they have determined that our solar system is painfully average in the methanol department.  In other words, we're not all that special and life still managed to appear on earth.  The implication here is there may be other solar systems out there with greater methanol concentrations that lend themselves more easily to the appearance of life than our own!

Sources

http://www.sciencedaily.com/releases/2011/11/111106142036.htm

http://www.esa.int/SPECIALS/Mars500/

http://www.sciencedaily.com/releases/2011/11/111103190356.htm

http://www.sciencedaily.com/releases/2011/11/111102190028.htm

Hydrostatic Equilibrium and the Sun

Abstract
We would like to know how the sun is being "supported".  We assume that this mechanism is hydrostatic equilibrium, but to be sure we work through the derivation.

Introduction
We know that the sun is somehow being prevented from gravitational contraction.  Our theory is that it is supported by hydrostatic equilibrium, which means that the internal pressure provides an opposing support force.  We calculate the gravitational force on a mass shell, the pressure required to balance it, and then derive the force equation for hydrostatic equilibrium.

Methods and Results
We first assume the Sun to be a spherical gas cloud with density ρ(r).  We consider a differential mass shell of this sphere with radius r.  We recall that the volume of a sphere is ⁴/₃πr³ and that the differential volume is its derivative. Then we get a differential mass dM:

We know the equation for universal gravitation:

Here we let M be the total mass enclosed by the mass shell and m be the differential mass element.  As a result, we get the differential gravitational force to be:

We know that pressure is equal to force divided by area.  So we can say:

Now dividing by dr on both sides of the equation we arrive at the equation of hydrostatic equilibrium:

Conclusions

We have derived from simple physical laws that the equation for hydrostatic equilibrium is a plausible explanation for the way the sun is supported.  A quick search shows that we are indeed correct.  Hooray!