?

How fast is our galaxy moving, and in which direction in celestial, ecliptic, or galactic coordinates?

e : eccentricity

M : mean anomaly

u : eccentric anomaly

Method #1.

u = M + e sin(M)

i = 1

REPEAT

i = i+1

ú = u

u = M + e sin(ú)

UNTIL |u−ú| < 1ᴇ-11

Method #2.

i = 1

u = M + (M + e sin(M) − M) / (1 − e cos(M))

REPEAT

i = i+1

ú = u

u = ú + (M + e sin(ú) − ú) / (1 − e cos(ú))

UNTIL |u−ú| < 1ᴇ-11

Method #3.

i = 1

u = M + e sin(M)

REPEAT

i = i+1

ú = u

u = M + e sin(u)

UNTIL i=3

REPEAT

i = i+1

ú = u

u = ú + (M + e sin(ú) − ú) / (1 − e cos(ú))

UNTIL |u−ú| < 1ᴇ-11

Method #4.

u = M + (e − e³/8 + e⁵/192) sin(M) + (e²/2 − e⁴/6) sin(2M) + (3e³/8 − 27e⁵/128) sin(3M) + (e⁴/3) sin(4M)

REPEAT

ú = u

A = u − e sin ú − M

B = 1 − e cos ú

C = e sin ú

D = e cos ú

E = −A / B

F = −A / (B + EC/2)

G = −A / (B + EC/2 + DF²/6)

u =  ú+G

UNTIL |u−ú|/u < 1ᴇ-11

I think that I solved Lambert's Problem on my own. But I'd like to see how Lagrange did it circa 1780. The problem is that of finding a transfer orbit between two points in space, given the heliocentric state vectors at each point just prior to departure (for the earlier point) and just after arrival (at the later point), and the time difference between arrival and departure.

Here's what I think.

Alexandria Ocasio-Cortez (Grand Wizard)

Ilhan Omar (Wanna-be Grand Wizard)

Ayanna Pressley (included for diversity)

Rashida Tlaib (included for diversity)

They're yummy, you must admit that.

Find the semimajor axis, the eccentricity, the inclination to the ecliptic, the longitude of the ascending node, the argument of the perihelion, and the time of perihelion passage for this asteroid.

A spaceship is initially in Earth's orbit, but is on the opposite side of the sun from Earth. Its captain wants to enter a transfer orbit, bound for Vesta, at 12h UT on 26 June 2017. The navigator does some trial runs on a computer and discovers an elliptical transfer orbit having its aphelion at Vesta upon arrival at 4h 45m 36.036s UT on 12 June 2018.

Check the navigator's work to ensure that an elliptical transfer orbit does exist for these times for departure and arrival. Show the elements of the transfer orbit and the delta-vees required for transfer orbit insertion (departure) and for matching velocity with Vesta at arrival.

Given...

That a spaceship orbits in the equatorial plane of an exoplanet , and an observer on the spaceship measures the following:

The exoplanet's angular equatorial diameter at periapsis

Dₑ₀ = 16.3060482°

The exoplanet's angular equatorial diameter at apoapsis

Dₑ₁ = 12.0634564°

The exoplanet's angular polar diameter at periapsis

Dᵨ₀ = 16.2567986°

The exoplanet's angular polar diameter at apoapsis

Dᵨ₁ = 12.0271323°

The spaceship's sidereal orbital period around the exoplanet

P = 108000 sec

The spaceship's altitude above the exoplanet's surface at periapsis

h₀ = 52568218 meters

Find...

The exoplanet's mass

The exoplanet's average density

The semimajor axis of the spaceship's orbit

The eccentricity of the spaceship's orbit

Present your answer as a ratio with the orbit's semimajor axis, and as a function of the orbit's eccentricity.

The early US satellite Explorer 1 was a long cylinder that was intended to rotate around its geometric axis of symmetry, i.e. the long axis running down the center of its length. Due to a small asymmetry in the distribution of mass, the satellite began to precess, and then started foolishly tumbling end-over-end.

There is a maximum for the ratio of length to diameter for the rotational stability of cylinders rotating around their geometric axis of symmetry. What is that maximum ratio?

Given the radiation flux from a source, you should be able to calculate the source's apparent magnitude, for the same spectral band. And likewise conversely. What is the equation for doing that?

A starship is in an elliptical orbit in the equatorial plane of an exoplanet.

When the starship is at periapsis, the exoplanet's equatorial angular diameter is 16.3060482° and its polar angular diameter is 16.2567986° .

When the starship is at apoapsis, the exoplanet's equatorial angular diameter is 12.0634564° and its polar angular diameter is 12.0271323°.

The period of the starship's orbit around the exoplanet is 30 hours.

The starship's minimum altitude above the exoplanet's surface is 52,568,218 meters.

What is the exoplanet's mass?

I personally don't know of any physical process that can't be modeled without need for tetration or higher order hyperoperations. In my own experience, exponentiation suffices. Does anyone else know about a physical process that can't be well-modeled unless tetration is used in the math?

Two bodies having a total mass M (i.e., M=M₁+M₂) are initially at rest, separated by a distance d, in vacuum, and isolated from all forces except their mutual gravitational attraction. Find the time elapsed from the initial time, t₀, to the instant, t₁, at which the separation is r₁, such that 0<r₁<d.

Why is the sky blue?

When will we land on Mars?

Did I just see an alien spaceship?

A starship is in an elliptical orbit in the equatorial plane of an exoplanet. The navigator makes the following observations:

The equatorial angular diameter at periapsis is 16.3060482° 

The equatorial angular diameter at apoapsis is 12.0634564° 

The polar angular diameter at periapsis is 16.2567986° 

The polar angular diameter at apoapsis is 12.0271323° 

The period of orbit is 108,000 seconds.

The altitude above surface of the exoplanet at periapsis is 52,568,218 meters.

Find the exoplanet's mass.

Find a general expression for the separation between two gravitationally bound masses in a elliptical orbits around a common center of mass have their average speed relative to each other. (Hint: it isn't the length of the semimajor axis.)

Two bodies having a total mass M (i.e., M = M₁ + M₂) are initially at rest, separated by a distance d, in vacuum, and isolated from all forces except their mutual gravitational attraction.

1. Find the time elapsed for the entire fall from the initial moment to contact.

2. Find the time elapsed from the initial moment to the moment at which the separation is r, such that 0<r<d.

3. Find the time elapsed beginning the instant at which the separation is d/2 and ending when the separation is d/3.

4. Find the fraction of the initial separation that is closed in half of the amount of time required for the entire fall to contact (assuming point masses).

A spaceship travelling in an unoccupied part of Earth's orbit changes its velocity in order to make a rendezvous with Vesta. Its position vector in heliocentric ecliptic coordinates for the departure burn is 

x₁ = −0.09273216409680376 AU

y₁ = +0.9790543154949219 AU

z₁ = 0.0

The position of the rocket at the time of arrival is

x₂ = −0.1329822455259632 AU

y₂ = −2.149578487312375 AU

z₂ = +0.08086760107675305 AU

Assume that the transfer orbit has its aphelion at the arrival position and calculate the eccentricity of the transfer orbit.