Physics

Gravitational Field

Kepler's laws, universal gravitation, field strength, gravitational potential energy, and satellites.

Kepler's 1st Law. Law of Orbits:
Planets move in elliptical orbits around the Sun, with the Sun at one of the two foci.
Kepler's first and second law
Kepler's 2nd Law. Law of Areas:
A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. Given any two points of the orbit:
r1⋅v1⋅sin⁡θ1=r2⋅v2⋅sin⁡θ2r_1 \cdot v_1 \cdot \sin \theta_1 = r_2 \cdot v_2 \cdot \sin \theta_2
At perihelion and aphelion,  heta1= heta2=90∘\ heta_1 = \ heta_2 = 90^\circ
rp⋅vp=ra⋅var_p \cdot v_p = r_a \cdot v_a
Kepler's 3rd Law. Law of Periods:
For a given planet, the square of its orbital period is directly proportional to the cube of its mean distance from the Sun.
T2=k⋅r3withk=4π2GMsunT^2 = k \cdot r^3 \quad \text{with} \quad k = \frac{4\pi^2}{G M_{sun}}
Where: r is the mean distance to the sun, which coincides with the semi-major axis of the ellipse.
Every point mass attracts every other point mass by a force acting along the line intersecting the two points. The force is proportional to the product of the two masses, and inversely proportional to the square of the distance between them.
Gravitational force between two masses
F⃗12=−Gm1⋅m2r2u⃗12(G=6.67⋅10−11N⋅m2kg2)\vec{\text{F}}_{12} = -G \frac{m_1 \cdot m_2}{r^2} \vec{\text{u}}_{12} \quad \left( G = 6.67 \cdot 10^{-11} \frac{\text{N} \cdot \text{m}^2}{\text{kg}^2} \right)
Where u⃗12\vec{\text{u}}_{12} is the unit vector from mass 1 to 2, and GG is the Universal Gravitational Constant.
Gravity is a central and conservative force.
A mass M creates a gravitational field around it, over which the field strength g⃗\vec{\text{g}}, or simply Gravitational Field, is defined as the force that would act on a unit of mass located at a point within that field.
g⃗=−GMr2u⃗in(Nkg=ms2)\vec{\text{g}} = -G \frac{M}{r^2} \vec{\text{u}} \quad \text{in} \quad \left( \frac{\text{N}}{\text{kg}} = \frac{\text{m}}{\text{s}^2} \right)
It corresponds to the acceleration of gravity that a body would experience at a point within the field.
The vector u⃗\vec{\text{u}} is a unit vector from the generating mass to the point where the field strength is calculated.
Superposition principle:
At a point under the influence of several fields, the gravitational field will be the result of the vector sum of the fields generated by each of the masses.
g⃗T=g⃗1+g⃗2+g⃗3+…\vec{\text{g}}_T = \vec{\text{g}}_1 + \vec{\text{g}}_2 + \vec{\text{g}}_3 + \dots
Gravitational Potential Energy
The gravitational potential energy of a mass at a point in space is the work done by a gravitational field to move the mass from that point to infinity. The origin of energy is that at which the force is zero, that is, the point r=∞r = \infty, where Ug=0U_g = 0.
Ug=−GM⋅mrU_g = -G \frac{M \cdot m}{r}
Gravitational Potential
The gravitational field can be associated with a scalar magnitude, gravitational potential V at a point, defined as the gravitational potential energy per unit mass placed at that point:
V=UgmorV=−GMrV = \frac{U_g}{m} \quad \text{or} \quad V = -G \frac{M}{r}
Superposition principle.
In a region of space under the influence of several gravitational potentials, the total potential at that point is the sum of the individual potentials.
VT=V1+V2+V3+…V_T = V_1 + V_2 + V_3 + \dots
Work done by a gravitational field
The work done by the field to move a particle from point A to point B, since the gravitational field is a conservative field, is the negative change in potential energy between these points. It can be expressed in different ways:
WA−B=−ΔUg=−(UgB−UgA)W_{A-B} = -\Delta U_g = -(U_{gB} - U_{gA})
WA−B=G⋅M⋅m(1rB−1rA)W_{A-B} = G \cdot M \cdot m \left( \frac{1}{r_B} - \frac{1}{r_A} \right)
WA−B=m(VA−VB)W_{A-B} = m(V_A - V_B)
Escape velocity: is the minimum vertical speed that must be imparted to a body from the surface of a planet of radius r, so that it escapes the influence of its gravitational field.
ve=2GMrv_e = \sqrt{\frac{2GM}{r}}
Orbital velocity: a satellite in its orbit balances the gravitational attraction and centrifugal forces:
GM⋅mr2=m⋅v2rG \frac{M \cdot m}{r^2} = \frac{m \cdot v^2}{r}
from which:
v=GMrT=2πr3GMv = \sqrt{\frac{GM}{r}} \quad T = 2\pi \sqrt{\frac{r^3}{GM}}
Where T is the period of revolution of the satellite.
Mechanical energy of orbital motion: is the sum of kinetic and potential energies:
E=−GM⋅mr+m⋅v22orE=−GM⋅m2rE = -G \frac{M \cdot m}{r} + \frac{m \cdot v^2}{2} \quad \text{or} \quad E = -G \frac{M \cdot m}{2r}