Just for clarification. I'm not enrolled and refamiliarizing myself with the content by going through a textbook called Orbital Mechanics for Engineering students (4th Ed). Anyways the problem is asking to show r <= mu/|epsilon| when epsilon < 0. Using the Vis Viva equation I showed assuming epsilon is negative, r is forced to be less than or equal to mu/|epsilon| otherwise |v| < 0 which is impossible. I checked the answer on Chegg (because I don't have a professor to get the answers from after i try and solve myself) and they started off from Vis Viva and getting r=2mu/v^2 but then just throw in |epsilon| =0.5mv^2 and divide by m to get specific energy but still called it |epsilon| I'm confused on how, based on Vis Viva, epsilon is total mechanical energy, and they just used the magnitude of epsilon being 0.5mv^2 which is total kinetic energy. Did they just assume r was going out to infinity? It feels weird to just throw that in there without a justification on why that's allowed. Personally, i liked my own reasoning through it, but the Chegg answer has me thinking I'm missing something on a more fundamental way.
I've been working through some questions and now I'm confused. GM/r represents the gravitiational potential energy lost bringing a unit mass object from infinity to r, but at the same time KE = 1/2mv^(2), so why is the equation not:
1/2v^(2)=GM/r
v=sqrt(2GM/r)
I have seen many practice questions involving answers being d orbital answers, such as sp2d, when does this occur and why?
I'm in year 12 (lower 6th) and doing an essay on the SchrΓΆdinger equation, and one of the topics is going to be it's applications but I'm finding typing that into Google is far too broad so I thought I'd come here to narrow it down.
So far I've done particle in a box with the time independant equation, and I'll probably describe how the electron wave equation exists as a standing wave around the nucleus with integer wavelengths, but I haven't found much linking it to orbitals in the way that I'd hoped. Do I need to look into the 3 dimensional version, or integrating over a sphere? Seems like everywhere I look involves slightly different equations and ways it's used.
Thanks for the help.
Here's my interactive simulation of the hydrogen orbital radial distribution function that I constructed using the formula I found online. The Generalized Laguerre and Associated Legendre polynomials are not automatically supported by Desmos, so I searched for them and merged the formulas together. The x gives the distance from the center of the atomic nucleus, while the y give the probability density for the electron to be located at that distance. Complex behaviors of the distribution that can arise, starting from the distribution for the p orbital, at varying points when the magnetic quantum number and the phi angle are not zero (likely to account for other dimensions) were rendered in terms of red and blue for real and imaginary parts. I wonder how you think about this representation.
Why can every single element be described using s,p,d and f orbitals? From my insubstantial understanding, an orbital is merely a region of space where there is a probability of 0.9 or some other constant of finding the electron. And we find the probability for that particular orbital from solving the wavefunction in time dependent Schrodinger equation.
However, different elements have different potentials, why then are the 1s orbitals (or any other orbitals) for any 2 elements the same? Why can we conveniently express electronic configuration for any element so systematically?
If the same equation is used for both problems, then i imaginne something has to be the same or similar for that to work. The only thing I could think of is the forces being the same or similar. Is my assumption correct?
So, I found this nifty page http://www.projectrho.com/public_html/rocket/engines.php explaining the basics of rocket performance and the trajectories you can use with various levels of delta v.
I've noticed many hard scifi enthusiasts always assume that every spaceship either uses very slow Hohmann transfers and painstakingly obey launch windows or some even lower energy trajectory or just do a full brachistrome trajectory ("flip and burn") using a high thrust. This is not the case, not even now (the Voyagers and New Horizons used a hyperbolic trajectory aided by gravity slingshots - a Hohmann transfer to Pluto would take 20 years - but due to the very low Isp of chemical rockets they had no chance of slowing down and entering orbit).
If you have a highly efficient, powerful drive (say, a gas core nuclear drive, or a fusion drive) that nevertheless does not quite have the Isp or thrust to maintain constant thrust, you can use very fast hyperbolic trajectories to go around the Solar system. It does not make sense in The Expanse that the time to arrive to Mars before the Epstein drive with a normal fusion rocket was 4 months - there is absolutely no reason why a fusion drive with a minimum Isp of about 10000 s. would use slow Hohmann transfer orbit unless you really wanted to save fuel.
An interplanetary society like the one portrayed in The Expense does not NEED the Epstein drive even if it is a nice thing to have - the first fusion spaceships that we will construct IRL (if we don't screw up our civilization before that) will certainly not be Epstein style torchships, but even a "weak" fusion drive would open up the Solar system. The Isp at which going on fast hyperbolic trajectories around the Solar system (without gravity assists and with the intention of braking) becomes worth it is about 10000 s or 1/100 of an Epstein drive, and even with less Isp your launch window widens until it becomes irrelevant. Best chemical rockets, unfortunately, have only a 450 s. Isp. We really need atomic (fission first, then fusion) and electric propulsion to open up the Solar system.
What's the biggest issue with relay networks? Losing sync! That is one satelite catching up to another. Annoying! If they're a bit more eccentric than each other or on a bit of an angle, that doesn't matter so long as their orbital periods are fine.
Now, as we all know (jk, wiki)
T = 2 PI sqrt(r^3/mu)
where T is the orbital period, and r is the semi major axis of the orbit and mu is the gravitational constant of the body we're orbiting, which you can look up (3.5316Γ10^12 for Kerbin).
How to find r though? It's just the average of the Apoapsis and Periapsis, plus the radius of kerbin, (Ap+Pe)/2 + 6e5. As it's the only thing that we can control in the equation, the TLDR: make all your satellites have equal Ap+Pe and they'll stay in sync.
Bonus: How much will my Sats go out of sync?
Calculate the above T for each of your satellites, and take the difference between them. Then
360*4*(T1 - T2)/(T1+T2)^2
will give the number of degrees it will go out per second, multiply by 3600*6 to get degrees per day. For example (follow along at home!):
Ap=2643365, Pe=2632488 => T1= 1.94802530E+04
Ap=2645892, Pe=2630129 => T2= 1.94810111E+04
T1-T2=0.758 sec difference.
360*4*(T1-T2)/(T1+T2)^2 * 3600 * 6 = 0.0155 deg/day
0.0155^o per day is around 7^o per year, not too bad! To really refine those numbers you can always thrust limit your propulsion to really get it right.
Edit: Forgot to add the radius of Kerbin in the axis calc.
I found an equation for the maximum eclipse period for a satellite around a planet in an elliptical orbit. I now cannot find where I got it. Does someone know how to derive it or know where I can reference to use the equation in some of my course work?
T = (P * R * cos^(2)(Thi/2) * Thi) / ( Pi * a^(2) sqrt(1- e^(2)))
Where T = the eclipse period
P = Orbital period
Thi is the angle from the centre of the planet to the point of the orbit where it enters and exits eclipse
a = semi-major axis
e = eccentricity
R = the radius at the beginning and end of the eclipse
Maybe im just over simplifying, i feel like it could work. Like centripetal force
In my University Chemistry class we were discussing molecular orbital theory, and how scientists use trial functions to determine the lowest energy configurations of molecular orbitals because the SchrΓΆdinger Equation for the molecular orbitals is impossible to solve. Is there a layman's (ish) explanation for why this is the case?
Thanks in advance.
Was tryna calculate a radius for a sattlite that has a orbital period of 2 hours and I keep screwing up. Please help me I am new to these equations. Thank you
Provided( 2 hour orbital period, Exact polar orbit) Thank you
Please comment if you need more information.
I want to create fantasy world setting that takes place on 4 seperate celestial bodies; two moons and two planets. The two planets are co-orbital and of about equal size and mass, and they are in orbit around co-orbital stars, each roughly the size of our sun, at about the same distance as our earth. To make this even crazier, I've decided that as these two planets orbit, a pair of moons zip along in figure 8s around them, ideally in such a pattern as to avoid collision. I know very little about astrophysics in general, and, avoiding whether or not this series of circumstances is actually possible, what sort of crazy things would result from such a setup? And is it possible for co-orbital planets to have a shared atmosphere in such a situation? Even if not, were it magically made so, what sort of impact would a shared atmosphere have? Thanks for your time and consideration!
I know how to derive the velocity of a circular orbit, but what about the equation for an elliptical orbit? I'm not well versed in ellipse geometry so can someone explain this?
I've just started reading Consider Phlebas and Banks has just described an orbital. While the concept isn't new to me, I have for the first time wondered about a couple of things.
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Is the graduation of gravity equivalent force from the outer rim (say 1G) to the the centre of the orbital linear or exponential?
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My main question, is there some kind of equation where you can plug in orbital diameter, rotary speed, and distance from centre and calculate the gravitational pull of that?
I guess if you wanted to figure out the gravity at the top of this 200 story building you'd actually subtract the height of this story from the radius and, ignoring that actual diameter of the orbital (where gravity is 1G) just work it out from there?
My mathematics is pretty poor so if any of this doesn't make sense please ask and I can try to elaborate.
Edit: it seems as if the proper terminology is centripetal force (thank you LaserHorse)
Edit 2: Answer (velocity^2 / radius) http://larryniven.net/physics/img4.shtml (thanks ronin1066!)
I'm trying to derive some equation(s) that describe one body orbiting around another. I realize that this has been done already, but I wanted to try it on my own with just first principles (Universal law of gravitation and F=ma). What I have so far:
A mass m orbiting mass M experiences an acceleration a = [;-\frac{MG\bar{p}}{|\bar{p}|^3} = \frac{d^2\bar{p}}{dt^2};]
This is my fundamental governing differential equation, from which I split the orthogonal vector p (position) into 2 different differential equations: [;r^3\frac{d^2r}{dt^2}+MGr=0;] and [;r^3\frac{d^2\theta}{dt^2}+MG\theta=0;]. The first one I can reformulate as [;r^2\frac{d^2r}{dt^2}=-MG;] and with a little thought, I thought of the solution [;r(t)=At^{2/3};] with [;A=\sqrt[3]{\frac{9MG}{2}};]
The problem with that, however, is that it has r(t) increasing without bound; I don't see how to allow for stable orbits with that. Now maybe that's because I need to solve [;r^2\frac{d^2r}{dt^2}=0;] or maybe I did something else wrong. In any case, where should I go from here?
I need to come up with an equation, and a friend suggested I post about it here. Seemed like just the thing to do, so here we go!
I'm trying to derive an equation that will help me draw a sine-wave-type graph, specifically the type that is used to show orbital flightpaths for the ISS and satellites and such, across a map in the Mercator projection. The input values should be the longitude and latitude of a point along the curve, and a 360-degree "heading" from that point.
Essentially, I want to be able to say "If I'm over New York City and angled at 20 degrees from north, what does my course across the world look like?"
Assume that the orbit is either instantaneous or at-pace with the Earth's rotation (which is to say, the path doesn't need to change with each orbit, a perfect circle/wave returning to the starting point is fine). Also assume a stable non-deteriorating orbit, and that we're working with standard longitude/latitude degree notation, from -180 to 180 and -90 to 90.
Can anyone help me out here? I'm sure there is an equation for this, it's complicated but fairly straightforward geometry, I'm just not fully up on my sines and cosines anymore...
Anyone have any ideas?
I've been trying to find an answer online but have been unsuccessful thus far.
Nuclei are tiny spheres, right? How do they make objects which are flat? How come one atom looks like electrons orbiting a nuclei, but millions of them make objects that we see not as electrons orbiting nuclei??
Does anyone know of an equation I can use to calculate the velocity required to circularly orbit a body at a known distance? For context I am trying to figure out the effect of time dilation at different distances from a black hole.
Whilst I'm at it, does anyone have an equation for finding time dilation around a black hole? Thanks!
