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Kepler’s Laws

The first law says:"The orbit of every planet is an ellipse with the sun at one of the foci."

where (r,ν) are heliocentricical polar coordinates for the planet, p is the semi-latus rectum, and ε is the eccentricity, which is greater than or equal to zero, and less than one. For ν=0 the planet is at the perihelion at minimum distance:

for ν=90°: r=p, and for ν=180° the planet is at the aphelion at maximum distance:

The semi-major axis is the arithmetic mean between rmin and rmax:

The semi-minor axis is the geometric mean between rmin and rmax:

and it is also the geometric mean between the semimajor axis and the semi latus rectum:

The second law: "A line joining a planet and the sun sweeps out equal areas during equal intervals of time."

The third law: "The squares of the orbital periods of planets are directly proportional to the cubes of the axes of the orbits." Thus, not only does the length of the orbit increase with distance, the orbital speed decreases, so that the increase of the orbital period is more than proportional.

P = orbital period of planet

a = semimajor axis of orbit

Hubble’s law

·  v is the recessional velocity, typically expressed in km/s.

·  H0 is Hubble's constant and corresponds to the value of H (often termed the Hubble parameter which is a value that is time dependent) in the Friedmann equations taken at the time of observation denoted by the subscript 0. This value is the same throughout the universe for a given comoving time.

·  D is the comoving proper distance from the galaxy to the observer, measured in megaparsecs (Mpc), in the 3-space defined by given cosmological time. (Recession velocity is just v = dD/dt).

Circular Motion

For motion in a circle of radius R, the circumference of the circle is C = 2π R. If the period for one rotation is T, the angular rate of rotation ω is:

The speed of the object traveling the circle is

The angle θ swept out in a time t is:

The acceleration due to change in the direction of the velocity is found by noticing that the velocity completely rotates direction in the same time T the object takes for one rotation. Thus, the velocity vector sweeps out a path of length 2π v every T seconds, or:

and is directed radially inward.

Speed of light~3×108 m/sà299,792,458 m/s

Light year~9.46×1012 km~9.46×1015 m

1 parsec= 3.26 light years; 1 light year=0.3067 parsec

solar radius~6.96×108 m~4.652×10-3 AU

solar mass~1.98892×1030 kg

CelsiusàKelvin=°C+273°

Solar temperature=?

-Solar luminosity=3.839×1026 watts or 3.839×1033 erg/s

-Distance modulusà log D=(m-M+5)/5; m=visual magnitude, M=absolute magnitude

-Absolute magnitudeà M=m+5+5log P; P=parallax in arc seconds, m=visual magnitude

-Brightness in inversely proportional to magnitude—a star of magnitude 1 is 100x brighter than a star of magnitude 6
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