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The pendulum fallacy is the belief that rockets would be passively stable with engines at the top, with the rocket "hanging" from them. The error lies in expecting gravity to pull the body of the rocket down while the engines pull it up. In reality, gravity acts on the body of the rocket and the engines equally, exerting no torque (except for ...


24

In the inverted pendulum problem: gravity exerts a vertical force on the pendulum, at the center of gravity the support of the pendulum (like the finger under the pencil) exerts a vertical force on the pendulum, at the bottom of it In a rocket: gravity is the same engines exert a force along the long axis of the rocket, where the engine is (which doesn't ...


11

Rockets produce thrust by ejecting reaction mass at some velocity. The fundamental quantities involved are mass flow rate and exhaust velocity, thrust is the consequence of these. It's no coincidence that specific impulse in units of velocity equals exhaust velocity, that's what specific impulse is. The exhaust velocity gives you the impulse per unit of ...


8

Your formulas are using a base-10 logarithm, not the natural logarithm. This causes all your delta-v values to be off by a factor of 2.3 In google sheets, you can supply the base of the logarithm as a second argument to the LOG function. For other systems, the identity $log_a(x) = \frac{log_b(x)}{log_b(a)}$ may be useful if you need to obtain a logarithm in ...


7

There are several ways to do this. The easiest and most straightforward is to break it into two sets by including velocity as a variable, and solve together. Instead of a single second order differential equation $$\ddot{\mathbf{r}} = -\frac{\mu}{r^3}\mathbf{r}$$ We can solve the following pair of first order differential equations in parallel $$\dot{\mathbf{...


7

One method of calculating the angle involves using the ellipse reflection law. Light from one focus reflects off the ellipse into the other focus. Thus in the picture below (by the author), the radial vector from the focus $F_1$ is reflected at $P$ onto the second focus $F_2$, forming a triangle whose third side is the line between the foci. Your flight ...


7

Discrete Fourier techniques introduce errors in their terms that track with the sinc function. Any target in the image will produce side lobes like those in the following graph. Strong reflections will create side lobes that have a higher amplitude than the background of the image. This is why the target appears to have a larger spatial extent than it ...


6

First, you appear to have the following misunderstanding of the solar sail force vectors: Tilt it at 45 degrees to make the thrust tangential Thrust is not tangential at 45 degrees. In fact, a solar sail always has thrust perpendicular to the sail, and can thus not achieve thrust perfectly tangential to the Sun, since the cross section would then be zero. ...


5

As $\Delta v$ is just change in velocity, we can just integrate the norm of the acceleration function over time: $$\Delta v = \int|\mathbf{a}(t)| dt$$ You're out of luck getting a closed form of that integral though. As far as analytical solutions goes, we can note that at $t = \frac{\pi}{2}$, all of $a_x$, $a_y$ and $a_z$ are maxed out, and hence $\Delta v &...


5

The Eq. 4.43 is in Battin's An Introduction to the Mathematics and Methods of Astrodynamics where he does not derive it. The derivation is outlined in his earlier book Astronautical Guidance (1964) on page 46. Starting from Kepler's equation for the two positions $E$ and $E_0$: $$M-M_0=E-E_0 -e(\sin E -\sin E_0) $$ make the substitution $\sin E = \sin[E_0 + (...


5

I'm sorry, I should have included more explanatory links in what I wrote earlier. When reading what I put together below, please keep in mind that each of these paragraphs is normally an entire grad school course in mathematics. I have attempted to clarify my meaning, but if I haven't succeeded, please note that it took me several years to understand all ...


5

Are bicomplex numbers including tessarines ever used in spaceflight (as an alternative to quaternions)? Without the parenthetical remark, the answer is "I don't know". But with that parenthetical expression, the answer is all-caps "NO". Unit quaternions are used in space exploration precisely because they map nicely to rotations in three ...


4

For coplanar orbits, a bi-elliptical transfer is more efficient than an Hohmann transfer when the ratio of the initial and final radii is greater than 15.58. When the ratio is less than 11.94, an Hohmann transfer is more efficient. (Thanks to notovny for correcting me.) A bielliptic transfer is effectively two subsequent Hohmann transfers. Section 6.3.2 of &...


4

Just to add to the existing established relativistic doppler shift is a continuous process. The equation is on the order of: $$1 + z=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$ Which contains no quantizing terms. The notion of redshift quantization (very neat! Thanks for telling me about this!) is only based on a few observations, whereas continuous redshift has ...


4

This is largely dependent on the radial velocity of the orbit of Mars, since its eccentricity is pretty large. Radial velocity is maximised at a distance of: $$r = \frac{r_P(2a - r_P)}{a}$$ Where $r_P$ is perihelion distance and $a$ is the semi-major axis. This part of the orbit is 156 days before perihelion. At this distance, the radial velocity of Mars ...


4

That parameter is known as the Effective Exhaust Velocity because... Sutton, 4th edition: When [the exit plane pressure is equal to the ambient pressure], the effective exhaust velocity is equal to the average actual exhaust velocity of the propellant gases...the effective exhaust velocity is usually close in value to the actual exhaust velocity.


4

There are several approaches, only differentiated by how big a budget you have and how accurate you want the answer - quite plausibly according to where you are in the programme / development proposal etc. First iteration: Just assume the thruster nozzle governs the mass flow rate and its completely expanded to ambient and that its discharge coefficient is 1....


4

There are really two questions here: Do there exist $n$-body systems with long-term stability? Can a third body (massive or not) be shown, a-priori, to be bounded or to escape—without resorting to numerical simulation? 1. Stability of $n$-body systems It is widely known that $n$-body systems are "chaotic" when $n>2$. However, this must be unpacked ...


4

If your ellipse is a circle, the Flight Path Angle is 0. You’re done. Otherwise, for an elliptical orbit, start with the polar equation that relates radial distance $r$, true anomaly $\theta$, semimajor axis $a$, and orbital eccentricity $e$: $$r=\frac{a(1-e^2)}{1+e\cos\theta}$$ Solving for $\theta$ gives us the following: $$\theta = \arccos\left({\frac{-...


4

The answer has to be 'not necessarily', because, in general, as you go along, you're free to adjust the solar sail angle and thus the trajectory. In addition, the trajectory need not lie in a single plane since the sail can produce out of plane forces. I posted an analysis in the comments yesterday for a weak solar sail and a shallow spiral orbit and ...


4

Yes, they're all just forms of the same relation that are specific to different values of eccentricity. This is one of the few topics treated in greater detail in Richard Battin's An Introduction to the Mathematics and Methods of Astrodynamics (chapter 4, pages 141–173) than in Vallado, including more biographical anecdotes about the mathematicians who ...


4

Matrix manipulation is mainly for systems of linear equations, and this is a system of nonlinear equations. You need to either change variables to form a linear system, or use a tool designed for nonlinear systems. If the system is overdetermined (has more equations than unknowns), or there is error in the measured values you plug into the equations, there ...


3

This question is really more of a general numerical analysis question, rather than a space exploration question (as there's no reason Kepler's equation is special in this regard), but I'll provide this guidance anyway: The divergence you are expecting comes from the fact that the $1-e \cos En$ term gets close to zero as the eccentricity approaches 1, meaning ...


3

The CSAR (coherent synthetic aperture radar) did not use very high frequencies and short wavelengths like 3 GHz (0.1 m) or 30 GHz (0.01 m) allowing small narrow beam directional antennas. Very low frequencies of 5 , 15 and 150 MHz and 60 , 20 and 2 m wavelength were used. These low frequencies were selected to image not only the lunar surface but also the ...


3

The flight path angle is simply the angle between the velocity vector and the vector perpendicular to the position vector. An easy way to visualise this: If the orbit was a circle, this angle would be zero. The angle is therefore due to the contribution of the inward/outward motion of the object away from the focal point. The semi major axis ($a$) and ...


3

...ensure the center of gravity is behind the point generating thrust Should read center of drag is behind the center of gravity relative to the direction of flight. In the air, rockets follow the same directional stability rules as their aircraft cousins, and indeed, arrows. Putting a crude stick on a rocket will increase its drag but enhance its ...


3

I believe I have worked out my own question using http://www.dept.aoe.vt.edu/~cdhall/courses/aoe4140/attde.pdf This uses the TRIAD algorithim in order to determine a rotation matrix between the body and inertial frames by knowing two vectors in both frames. Usefully, it also weights one as being more accurate than the other which in this case is likely to be ...


2

Assuming I have understood the constraints correctly, you have a solar sail in a (very gentle) spiral trajectory inwards, and want to bleed orbital energy at the highest possible rate. Considering edge cases, this is not always the optimal way of reducing transfer time. Imagine for instance falling straight down towards the sun, with no perpendicular ...


2

The calculation uses the following model for "total propulsive delta-v": $$\Delta v_{total} = \Delta v_{spacecraft} + \Delta v_{launcher}$$ Here, $\Delta v_{spacecraft}$ is what propulsive capabilities the probe has by itself after leaving the Earth system entirely, and is presumed to be a known value that can be looked up. $\Delta v_{launcher}$ is ...


2

There are some combinations of a large eccentricity and a bad guess for the initial value of the eccentric anomaly where a Newton-Raphson iterator of Kepler's equation appears to diverge. An initial guess of eccentric anomaly being equal to the mean anomaly seems completely reasonable and works quite nicely when eccentricity is not large. But when ...


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