Tsiolkovsky rocket equation

The maximum change of velocity of the vehicle, \( \Delta v \) (with no external forces acting) is:

\[ \Delta v = v_e \ln \frac{m_0}{m_f} = I_{sp}g_0 \ln \frac{m_0}{m_f} \]

where:

• \( v_e = I_{sp}g_0 \) is the effective exhaust velocity
  • \( I_{sp} \) is the specific impulse in dimension of time
  • \( g_0 \) is standard gravity
• \( m_0 \) is the initial total mass, including propellant, a.k.a. wet mass
• \( m_f \) is the final total mass without propellant, a.k.a. dry mass

Given the effective exhaust velocity determined by the rocket motor's design, the desired delta-v (e.g. orbital speed or escape velocity), and a given dry mass \( m_f \), the equation can be solved for the required propellant mass \( m_0 - m_f \):

\[ m_0 = m_f e^{\frac{\Delta v}{v_e}} \]

The necessary wet mass grows exponentially with the desired delta-v.

Hyperbolic excess velocity

Escape speed

Escape speed at a distance d from the center of a spherically symmetric primary body (such as a star or a planet) with mass \( M \) is given by the formula

\[ v_{e} = \sqrt{\frac{2\mu}{r}} \]

Specific orbital energy

In the gravitational two-body problem, the specific orbital energy \( \epsilon \) (or vis-viva energy) of two orbiting bodies is the constant sum of their mutual potential energy \( \epsilon_p \) and their total kinetic energy \( \epsilon_k \), divided by the reduced mass. According to the orbital energy conservation equation (also referred to as vis-viva equation), it does not vary with time:

\[ \epsilon = \epsilon_k + \epsilon_p = \frac{v^2}{2} - \frac{\mu}{r} \]

Equation forms for different orbits:

Hyperbolic excess velocity

When given an initial speed \( v_d \) greater than the escape speed \( v_e \), the object will asymptotically approach the hyperbolic excess speed \( v_{\infty} \), satisfying the equation:

\[ v_{\infty}^2 = v_d^2 - v_e^2 \]

where \( v_d \) is the departure velocity.

Sphere of influence (SOI)

Sphere around each planet inside which the motion of a spacecraft is considered to be two-body Keplerian.
The radius of the sphere of influence \( r_{SOI} \) has been determined by Laplace as:

\[ r_{SOI} \approx a \left( \frac{m}{M} \right)^{\frac{2}{5}} \]

where:

• \( a \) is the semimajor axis of the planet's orbit around the Sun.
• \( m \) and \( M \) are the masses of the planet and the Sun, respectively.

Catch up rate for nearby orbits

For circular orbits with \(\Delta r \ll r\): \[ \Delta X \approx 3\pi \Delta r \]

If two objects are on the same local vertical at some point, after one full orbit, the lower object will have moved forward by a distance equal to ten times the difference in attitude.

For circular / close elliptical orbits with \(|r - a| \ll r\): \[ \Delta X \approx 3\pi (r - a) \]

Change of orbital plane at equator crossing

\[ \Delta V = 2V_i \sin \left( \frac{\alpha}{2} \right) \]

This is generally costly; in LEO, a 10° change of orbital plane costs a ΔV of close to 1.4 km/s, and consequently a large quantity of propellant!

Hohmann transfer orbit

Launching to Mars requires a spacecraft to travel in an elliptical orbit about the sun such that the spacecraft and Mars will arrive in the same place at the same time.
The most energy efficient orbit of this type is called the Hohmann transfer, in which the spacecraft will travel half of one orbit about the sun, leaving Earth at the orbit’s perihelion and arriving at Mars (or any outer planet) at the orbit's aphelion.

The red line indicates the orbit of Mars, the blue line indicates the orbit of Earth, and the grey line indicates the path a spacecraft takes from Earth to Mars when launched on a Hohmann transfer path. Courtesy NASA/JPL-Caltech.

With the vis-viva equation, the delta-v (Δv) required for the Hohmann transfer can be computed as follows, under the assumption of instantaneous impulses:

\[ \Delta V_{1} = \sqrt{\left(\frac{2\mu r_{2}}{r_{1}(r_{1} + r_{2})}\right)} - \sqrt{\frac{\mu}{r_{1}}} \] \[ \Delta V_{2} = \sqrt{\frac{\mu}{r_{2}}} - \sqrt{\left(\frac{2\mu r_{1}}{r_{2}(r_{1} + r_{2})}\right)} \]

While the Hohmann maneuver often uses the lowest possible amount of impulse (which consumes a proportional amount of delta-v, and hence propellant) to accomplish the transfer, it requires a relatively longer travel time than higher-impulse transfers.

Case of small ΔV, For LEO

\[ \Delta r \approx 3.5\Delta V \]

*\( \Delta r \) in km and \( \Delta V \) in \( \frac{m}{s} \).

Vis-viva equation (orbital-energy-invariance law)

For any Keplerian orbit (conical: elliptic, parabolic, hyperbolic, or radial)

\[ v^2 = GM \left({ 2 \over r} - {1 \over a}\right) \]

where:

• \( v \) is the relative speed of the two bodies
• \( r \) is the distance between the two bodies' centers of mass
• \( a \) is the length of the semi-major axis (\( a > 0 \) for ellipses, \( a = \infty \) or \( \frac{1}{a} = 0 \) for parabolas, and \( a < 0 \) for hyperbolas)
• \( G \) is the gravitational constant
• \( M \) is the mass of the central body

The product of \( G \cdot M \) can also be expressed as the standard gravitational parameter using the Greek letter \( \mu \).

Virial theorem

For bound systems in stable orbits, the average kinetic energy is equal to half the absolute value of the average potential energy

\[ K = -\frac{1}{2} U \]

Conservation of Mechanical Energy

\[ E_{\text{mech}} = K + U_g = \text{constant} \]

for an isolated system, i.e. no external work.

Fubini's Theorem

Theorem

If \( f \) is continuous on the rectangle

\[ R = [a, b] \times [c, d] = \{(x, y) | a \leq x \leq b, c \leq y \leq d\} \] , then \[ \int\int_R f(x, y) \, dA = \int_a^b \int_c^d f(x, y) \, dy \, dx = \int_c^d \int_a^b f(x, y) \, dx \, dy \]

Critical points

Definition:

A point \( p \) in the domain of a function \( f \) of \( n \) variables is called a critical point of \( f \) if

• \( f_i( \)\( p \)\( ) = 0 \) for \( i = 1, \ldots, n \) or
• one of the partial derivatives fails to exist at \( p \).

Theorem:

Suppose \( f( \)\( x \)\( ) \) has a local minimum or local maximum at the point \( p \). Then \( p \) is a critical point of \( f \).

The Gradient

From... \[ D_{u}f(\mathbf{P}) = \nabla f(\mathbf{P}) \cdot \mathbf{u} = |\nabla f(\mathbf{P})| \cdot \cos(\theta) \]

Here are some key properties of the gradient \( \nabla f \):

1. Direction of Steepest Ascent:

\( \nabla f \) indicates the direction in which this derivative is maximized, since the dot product \( \nabla f \cdot \mathbf{u} \) is maximized when \( \nabla f \) and \( \mathbf{u} \) are parallel. Here, \[ \mathbf{u} = \frac{\nabla f}{\|\nabla f\|} \] If we want to maximize \( D_{u}f \), we let \( \mathbf{u} \) be parallel to \( \nabla f \), and in this case \[ D_{u}f = \|\nabla f\| \] where the magnitude \( \|\nabla f\| \) indicates the maximum rate of change.

2. Orthogonal to Level Curves/Surfaces:

For a function \( f(x, y) \) at point \( P = (x_0, y_0) \), say \( \nabla f(x_0, y_0) = (a, b) \). The equation of the tangent line to the level curve \( f(x, y) = c \) at \( (x_0, y_0) \) is given by:

here, \( k = a \cdot x_0 + b \cdot y_0 \), by plugging in \( (x_0, y_0) \).

Directional derivatives

To calculate the directional derivative of a function \( f(x, y, z) \) at a point \( (x_0, y_0, z_0) \) in the direction of a vector \( \mathbf{u} = (v_1, v_2, v_3) \),

\[ D_{\mathbf{u}}f = f_x v_1 + f_y v_2 + f_z v_3 = \left( f_x, f_y, f_z \right) \cdot (v_1, v_2, v_3) = \nabla f \cdot \mathbf{u} \]

Here, \( \nabla f \), known as the gradient vector (field) of \( f \), is a vector consisting of the partial derivatives of \( f \). It is defined as

\[ \nabla f = \left( f_x, f_y, f_z \right) = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right) \]

And, the gradient vector \( \nabla f \) at the point \( (a_0, a_1, a_2) \) is given by

\[ \nabla f(a_0, a_1, a_2) = \left( f_x(a_0, a_1, a_2), f_y(a_0, a_1, a_2), f_z(a_0, a_1, a_2) \right) \]

In another sense,

\[ D_{u}f(\mathbf{P}) = \nabla f(\mathbf{P}) \cdot \mathbf{u} = |\nabla f(\mathbf{P})| \cdot \cos(\theta) \]

First order Linear differential equations

A first-order linear differential equation is one that can be put into the form

\[ \frac{dy}{dx} + P(x)y = Q(x) \]

Remark: We call the equation above the standard form
Example:

\[ y' - 2e^{x}sin(x)y = x^{2}e^{x} \]

Algorithm to solve 1st order Linear differential equations:

1. Write in standard form: \( \frac{dy}{dx} + P(x)y = Q(x) \)
2. Compute integrating factor: \( I(x) = e^{\int P(x)\,dx} \)
3. Multiply by \( I(x) \): \( [I(x)y]' = I(x)y' + I(x)P(x)y = I(x)Q(x) \)
4. Integrate: \( I(x)y = \int I(x)Q(x)\,dx \)
5. Divide by \( I(x) \): \( y(x) = \frac{1}{I(x)}\int I(x)Q(x)\,dx \)

Separable differential equations

A (first-order) separable differential equation is a first-order differential equation in which the expression for \( y' = \frac{dx}{dy} \) can be factored as a function of \( x \) time a function of \( y \).

In other words, it can be written in the form

\[ \frac{dy}{dx} = f(x)g(y) = \frac{f(x)}{h(y)} \]

Algorithm to solve separable differential equations:

1. write DE as \( \frac{dy}{dx} = \frac{f(x)}{h(y)} \)
2. Separate variables: \( h(y)dy = f(x)dx \)
3. Find anti-derivatives: \( H(y) = \int h(y)\,dy = \int f(x)\,dx = F(x) + C \)
4. Implicit solution: \( H(y) = F(x) + C \) with \( C \in \mathbb{R} \)
5. *Explicit solution: \( y(x) = H^{-1}(F(x) + C) \) with \( C \in \mathbb{R} \)
6. *if possible

Integration by Parts

\[ \int_{a}^{b} f(x)g(x)\,dx = \left. F(x)g(x) \right|_{a}^{b} - \int_{a}^{b} F(x)g'(x)\,dx \]

where \( F(x) \) is an anti-derivative of the function \( f(x) \).

\[ \int u\,dv = uv - \int v\,du \]

Here, \( u \) takes the role of \( g(x) \), while \( v \) takes the role of \( F(x) \). In fact, these variables \( u \) and \( v \) are still functions of \( x \), even though we do not make this explicit.

Method of substitution

If \( u = g(x) \), then \( du = g'(x)dx \). Therefore,

\[ \int f(g(x))g'(x)\,dx = \int f(u)\,du \]

For definite integrals,

\[ \int_{a}^{b} f(g(x))g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du \]

Fundamental theorem of calculus

If the function \( f(x) \) is intergrable* over the interval \( [a, b] \), then

Part 1

\[ \int_{a}^{b} f(x)\,dx = F(b) - F(a) \]

where \( F(x) \) is any anti-derivative of \( f(x) \).
intergrable*: A function \( F(x) \) is an antiderivative of \( f(x) \) if \( \frac{dF}{dx} = f(x) \).

Part 2

\[ F(x) = \int_{a}^{x} f(t)\,dt \]

is an anti-derivative of \( f(x) \), i.e. \( F'(x) = f(x) \).

Squeeze theorem

Suppose \( f(x) \), \( g(x) \) and \( h(x) \) are three functions such that

\[ g(x) \leq f(x) \leq h(x) \]

for \( x \) greater than some value \( M \) and suppose

\[ \lim_{{x \to \infty}} g(x) = a = \lim_{{x \to \infty}} h(x) \]

for some value \( a \) (horizontal asymptote). Then

\[ \lim_{{x \to \infty}} f(x) = a \]

Taylor's inequality theorem

Let \( f \) be a real-valued function that is \( (n+1) \)-times differentiable on an interval containing \( a \). Let \( T_n(x) \) be the \( n \)-th Taylor polynomial centered at \( a \) for \( f \), given by

Then, for any \( x \) in the interval containing \( a \), there exists a number \( c \) and \( M \) an upper bound for the absolute value of the \( (n+1) \)-th derivative of \( f \),

\[ |f^{(n+1)}(c)| \leq M \]

such that the error \( R_n(x) \) between the actual value of \( f(x) \) and the value predicted by the Taylor polynomial satisfies

\[ |f(x) - T_n(x)| \leq \frac{{M}}{{(n+1)!}}|x-a|^{n+1} \]

This inequality provides an upper bound on the error between the function \( f(x) \) and its Taylor polynomial approximation \( T_n(x) \), making it a useful tool for estimating the accuracy of the approximation.

Taylor Polynomials

The \( n \)th order Taylor polynomial of \( f \) at the point \( a \) is

\[ T_n(x) = \sum_{k=0}^{n} \frac{1}{k!}{f^{(k)}(a)}(x-a)^k \]

Basic property: \( f^{(k)}(a) = T_n^{(k)}(a) \) for \( k \) = 0, 1,..., n. The derivative of the \( k \)th term in \( T_n \)is \( f^{(k)}(a) \).

Differentials (a.k.a. very small differences)

Given a function \( y = f(x) \) the differential \( dy = df \) of \( f \) is given by

\[ dy = df = f'(x) dx \]

Linearization

\[ L(x) = f'(a)(x - a) + f(a) \]

The linear function whose graph is the tangent line at \( x = a \) is called the linearization of \( f \) at \( a \).

Derivatives of Inverse Trigonometric Functions

Standard Derivatives

Computation Rules

Tangent line

Given a function \( f(x) \) the tangent line to the graph of \( f \) at the point \( a \) is given by

\[ y = f'(a)(x - a) + f(a) \]