What is Circular orbit: Definition and 90 Discussions
A circular orbit is an orbit with a fixed distance around the barycenter; that is, in the shape of a circle.
Listed below is a circular orbit in astrodynamics or celestial mechanics under standard assumptions. Here the centripetal force is the gravitational force, and the axis mentioned above is the line through the center of the central mass perpendicular to the plane of motion.
In this case, not only the distance, but also the speed, angular speed, potential and kinetic energy are constant. There is no periapsis or apoapsis. This orbit has no radial version.
Schutz finds that the orbital period for a circular orbit in Schwarzschild is
$$ P = 2 \pi \sqrt {\frac { r^3} {M} }$$
He gets this from
$$ \frac {dt} {d\phi} = \frac {dt / d\tau} {d\phi/d\tau} $$
Where previously he had ## \frac {d\phi}{d\tau} = \tilde L / r^2## and ## \frac {dt}{d\tau} =...
Problem: a particle of mass m is in a circular orbit around a planet at a distance R from the center. The planet mass is M and it's radius is R_0.
What is the tangential impulse that will cause the particle to brush against the back of the planet? Describe the orbit.
The attempt at solution...
Hi :) This is a problem from David Tong's Classical Dynamics course, found here: http://www.damtp.cam.ac.uk/user/tong/dynamics.html. In particular it is problem 6ii in the first problem sheet.
Firstly we can see the lagrangian is ##L = \frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2+\dot{z}^2) -...
a. V=-GM/r
V=-6.67*10^-11*6.0 x 10^24/6.4 x 10^6
V grav = -62531250 ~ -62.5M Jkg^-1
b. To find the gravitational potential 200 km above the surface of the Earth;
r=6.4 x 10^6 +2*10^5 m=6.6*10^6
V grav=-6.67*10^-11*6.0 x 10^24/6.6*10^6
V grav= -60636363 ~ -60.6 M Jkg^-1
Can I check that it is...
Below are equations/formulas/text from
https://en.wikipedia.org/wiki/Schwarzschild_geodesics
https://hepweb.ucsd.edu/ph110b/110b_notes/node75.html
I apologize for not remembering the source for the "v=" equation, or for my inability to find it again.
For a circular orbit, the distance r and...
I know this problem can be solved using energy conservation, but I tried another method that I don't know is correct or not, but yielded a similar result to what my classmates got:
$$F_{C}=F_{G}\Rightarrow \frac{mv^{2}}{r}=\frac{GMm}{r^2}$$
$$\frac{v^2}{r}=\frac{Gm}{r^2}\Rightarrow...
In a circular orbit, the 4-velocity is given by (I have already normalized it)
$$
u^{\mu} = \left(1-\frac{3M}{r}\right)^{-\frac{1}{2}} (1,0,0,\Omega)
$$Now, taking the covariant derivative, the only non vanishing term will be
$$
a^{1} = \Gamma^{1}_{00}u^{0}u^{0} + \Gamma^{1}_{33}u^{3}u^{3}
$$...
Distance is d=1/0.07 = 14.3 parsecs
The Doppler shift of one star is, Δλ = 512 - 512.04 = -0.04
So the radical of the velocity of the star is = (-0.04/512) x (3.00 x 10^5 km/s) = 23.4km/s which is the same for both stars because they have the same mass.
This is as far as I've got.
Homework Statement
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To take the ##lim J \to \infty ##, what are the two roots of ##r_c## in this case...
So I believe it says' ##J## big enough it had 2 solutions' is basically saying just avoiding the imaginary solutions i.e. ## J^4 \geq 12G^2M^2J^2 ## (equality for one route obvs)...
Homework Statement
Consider a central force is attractive but which passes through the force center. In other words, consider an orbit of radius a which is centered at (a,0), with the force center at the origin
c.) Suppose the speed at the apogee is v0 Find the oribital speed v as a function of...
Homework Statement
A satellite of mass m is in a circular orbit of radius R about a star of mass M. The star ejects 1% of its mass by means of a spherically symmetric wind which removes the mass to a large distance. What are the new nearest and furthest distances of the satellite’s orbit around...
What If the velocity of particle moving in a circular orbit has increased , would the particle be no longer in circular orbit or it would go in an orbit with bigger radius?
The potential energy of a particle of mass $m$ is $U(r)= k/r + c/3r^3$ where $k<0$ and $c$ is very small. Find the angular velocity $\omega$ in a circular orbit about this orbit and the angular frequency $\omega'$ of small radial oscillation about this circular orbit. Hence show that a nearly...
Homework Statement
Homework Equations
The Attempt at a Solution
Is the particle moving under the influence of a single central force ?
Since the force always acts towards the center , work done by the force is zero . Energy is conserved . Potential energy at a particular radius can be...
Homework Statement
My title was supposed to say "Finding the radius of the satellites circular orbit" but I can't seem to edit it.
<< Mentor Note -- Title fixed for you >>
A 500 kg satellite experiences a gravitational force of 3000 N, while moving in a circular orbit around the earth.
a)...
Homework Statement
A satellite is in a circular orbit (radius R) around a planet of mass M. To change the satellite's orbit the engines fire and its speed is suddenly doubled. The engines fire for a very short time. Determine the length of the semi-major axis of the new orbit.
Homework...
Homework Statement
Homework EquationsThe Attempt at a Solution
[/B]Assuming that the relation ## v = \omega r ## is valid in relativistic calculation.
As the speed is greater for higher kinetic energy, ## \omega_2 ## is greater.
This is shown by only (d). So, I think (d) should...
Homework Statement
An object with mass m is attached to a string with initial length R, and moves on a frictionless table in a circular orbit with center C as shown in the figure. The string is also attached to the center, but its length is adjustable during the motion. The object initially...
Homework Statement
A satellite is in a circular orbit passing over the North and South geographical poles as it orbits the Earth. It has a mass of 2200kg and its orbit height is 870km above the Earth's surface. What is the change in momentum of the satellite from when it passes over the...
Hello,
I will be thankful if you could explain what appears to me as a paradox.
We know that a satellite on a circular orbit, let say around the earth, has a uniform speed given by v=√(GM/r0).
Now I would like to accelerate the satellite by keeping it on the same circular orbit r0. The only...
Hi! first time poster here.
I'm making an orbital simulation and I am having a problem with one minor detail.
The gravity is working great, and I've programmed it using this formula:
A force vector is applied = DirectionOfCentralBodyNormalized * ((GravConstant * centralbodymass *...
Homework Statement
If a proton with a kinetic energy of 6 MeV is traveling in a particle accelerator in a circular orbit of radius 0.75m, what fraction of its energy does it radiate per second?
m = 1.67 * 10^-27
epsilon_0 = 8.854 * 10^-12
c = 3 * 10^8
Homework Equations
dE/dt = (q^2 a^2) /...
Homework Statement
At a circular orbit and at an elliptical orbit when do I use 1/2mv^2 instead of the kinetic equation from which I derived from F=GMm/r^2 which is Ek = GMm/2r
Homework Equations
F=GMm/r^2, Ek = GMm/2r,
Ek = 1/2mv^2
The Attempt at a Solution
For instance, when I tried to...
So, it seems there's a new twin paradox thread every day, but I don't think I've seen this particular situation looked at.
The two twins move towards each other each in a circular orbit, and at one point they get close enough to each other that they can compare clocks directly (but their...
I finally found a result I believe for the the asymptotic metric (valid for large r) of a pair of bodies in a circular orbit emitting gravitational waves. I use spherical coordinates, ##[t, r, \theta, \phi]##.
If we let the linearized metric ##g_{\mu\nu}## be equal to the sum of a flat metric...
Homework Statement
The potential energy of a particle of mass m is V(r) = k/r + c/3r^3 where k<0 and c is a small constant. Find the angular velocity \omega in a circular orbit of radius a and the angular frequency \omega' of small radial oscillations about this circular orbit. Hence show...
Homework Statement
A satellite with mass 848 kg is in a circular orbit with an orbital speed of 9640 m/s around the earth. What is the new orbital speed after friction from the earth’s upper atmosphere has done -7.50·109 J of work on the satellite? Does the speed increase or decrease?
Homework...
The full power series for the Schwarzschild portion of perihelion shift is given in Mathpages as:
where L = a(1-\epsilon^2), a the semi-minor axis and \epsilon the eccentricity. This implies that as \epsilon tends to zero, the perihelion shift tends to a non-vanishing 6\pi m/a + some much...
Two satellites A and B move around Earth in a circular orbit. The mass of B is twice the mass of A then
I agree that kinetic energy of B is greater than that of A. But what I couldn't understand was that why are speeds of A and B equal as given in the book.
Shouldn't they differ according to the...
Homework Statement
A 15 kg satellite has a circular orbit with a period of 4.4 h and a radius of 3.5 × 106 m around a planet of unknown mass. If the magnitude of the gravitational acceleration on the surface of the planet is 1.2 m/s2, what is the radius of the planet?
Homework EquationsThe...
Homework Statement
Attached.
Homework Equations
The Attempt at a Solution
Hi. This question has me really puzzled; I simply don't know where to start with this one and thus am not sure of any relevant equations. It seems to be a problem about small perturbations to the circular...
I am self studying Kleppner and Kolenkow's an Introduction to mechanics. But i have one doubt about how they got into the equation no 3 of the example problem 9.3 in Central Force Motion.
Please clarify my doubt.
Homework Statement
(a) Find the proper time in the rest frame of particle
(b) Find the proper time in the laboratory frame
(c) Find the proper time in a photon that travels from A to B in time P
Homework EquationsThe Attempt at a Solution
Part(a)
[/B]
The metric is given by:
ds^2 =...
Homework Statement
In a classical model of a multi-electron atom, electrons are assumed to move in a modified electrostatic potential $V(r)$, given by;
$$V(r)=\dfrac{-k}{r}e^{-r/a}$$
Show that the effective potential is ;
$$V_e(r)=\dfrac{J^2}{2mr^2}+\dfrac{-k}{r}e^{-r/a}$$
Then show that...
Homework Statement What is the acceleration due to gravity at an altitude of 1.00 x 10^6 above the Earth's surface, given that the radius of the Earth is 6.38 x 10^6 m?Homework Equations
Using g = GMm/r^2
Fc = mv^2/r
The Attempt at a Solution
Fc = mv^2/r
How can I solve this equation if I am...
In uniform circular motion, (eg, a mass on the end of a string moving in a horizontal circle) centripetal force is the only thing causing acceleration. we have the kinematic relationship V=RW
or velocity is proportional to radius. I.e a bigger radius means greater linear speed?
For the...
Homework Statement
Show that the stability condition for a circular orbit of radius a, i.e.
f(a) + \frac{a}{3} (\frac{df}{dr})_{r=a} < 0
is equivalent to the condition
\frac{d^2V(r)}{dr^2} > 0
for r=a where V(r) is the effective potential given by
V(r) = U(r) +...
Homework Statement
I found this in Binney's text, pg 154 where he described the radial probability density ##P_{(r)} \propto r^2 u_L##
Homework Equations
The Attempt at a Solution
Isn't the radial probability density simply the square of the normalized wavefunction...
If an object is orbiting on a circular time-like geodesic path around a mass then the Wikipedia claims that the first component of its four-velocity is given by
\frac{dt}{d\tau} = \frac{1}{\sqrt{1-\frac{3}{2}\cdot \frac{r_0}{r}}}
where r_0 is the Schwarzschild radius.
Is this right and how...
Homework Statement
Two particles pigs of mass ##m## and ##M## undergo uniform circular motion about each other at a separation of ##R## under the influence of an attractive force ##F##. The angular velocity is ##\omega## radians per second. Show that R = \frac{F}{\omega^2}\left( \frac{1}{m} +...
Homework Statement
A particle of unknown mass moves in a circular orbit of radius R under the influence of a central force centred at some point inside the orbit. The minimum and maximum speeds of the particle are vmin and vmax respectively. Find the orbital period T in terms of these speeds...
The space shuttle is moving in a circular orbit with a speed of 7.8km/s and a period of 87min. In order to return Earth, the shuttle fires its retro engines opposite to its direction of motion. The engine provide a deceleration of 6m/s^2 that is constant in magnitude and direction. What is the...
Homework Statement
What is the radius of a circular orbit about the Earth where the acceleration is 0.1g?
Homework Equations
g=9.8m/s^2
F=(Gmm)/r^2
Radius of Earth= 6.38x103 km
Mass of Earth= 5.98x10^24 m/s
The Attempt at a Solution
I said 1/(2^x)=1/10 then did log2(10)=x and solved...
Let be a photon in a circular orbit (r=3M). I want to know the period measured by a stationary observer at this radius.
Because we are working with a photon we cannot 'talk' about proper time, that's why I don't understand how can I calculate this.
If I had a massive particle I could do...
Homework Statement
How much energy is required to move this satellite to a circular orbit with a radius of 2.50×104 miles?
I've found the answer to Part A of this problem to be 1.59*10^10, if this helps solve the above^^
(Find the kinetic energy of a 1.80×103 kg satellite in a circular...