Help With Simple Orbital Modeling

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Summary:

I would like to model the motion of a single celestial body with initial velocity v a distance r from a mass fixed in place . For the purpose of the question let's just say that the mass appeared r meters away from our celestial body perpendicular to its velocity. There are no outside forces or other orbits acting on these bodies and they will act as spherical point-masses. Where should I start with this modeling? Which equations will be most relevant to me? Should this be done on a polar plane?

Main Question or Discussion Point

I have yet to decide on values for the mass of the fixed object, M, the mass of the moving celestial body, m, the initial velocity, v, and the distance between the two objects, r. I will most definitely decide on a larger mass M because I would like the celestial body to spiral in towards the mass rather than be deflected slightly or fall into a stable orbit. I have attached an image illustrating the concept of my problem. This is for a calculus class, so I plan to preform this task primarily through the use of calculus.
Thank you for your time,
William B
problem.JPG
 

Answers and Replies

Janus
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Without an outside force acting on your object (other than gravity), it can't "spiral" in. There are only two possible types of trajectory for such an object starting with an initial velocity and acting only in response to gravity. An open trajectory ( either hyperbolic or parabolic) or a closed orbit (circular or elliptical).

If v is equal to or greater than ## \sqrt{2GM/r}##, then you will get the open trajectory, if it is less, then you get the closed trajectory. ( The direction vector of v doesn't matter)
 
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Did you take any Classical Mechanic course ?
 
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Without an outside force acting on your object (other than gravity), it can't "spiral" in. There are only two possible types of trajectory for such an object starting with an initial velocity and acting only in response to gravity. An open trajectory ( either hyperbolic or parabolic) or a closed orbit (circular or elliptical).

If v is equal to or greater than ## \sqrt{2GM/r}##, then you will get the open trajectory, if it is less, then you get the closed trajectory. ( The direction vector of v doesn't matter)
Shouldnt we include also ##V_{eff} = \frac{l^2}{2\mu r^2}## Such that ##V(r) = -k/r + \frac{l^2}{2\mu r^2}## and decide the motion wrt to this equation.
 
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How To Reduce Keplerian Orbital Elements and a time to Heliocentric Position and Velocity

Definitions

a : semimajor axis of orbit, positive for elliptical orbits, negative for hyperbolic orbits

e : eccentricity of orbit, between 0 and 1 for elliptical orbits, greater than 1 for hyperbolic orbits

i : inclination of the orbital plane to the ecliptic plane

Ω : longitude of the ascending node; the angle, subtended at the sun, measured in the ecliptic, from the vernal equinox to the place where the orbital plane intersects the ecliptic with the motion of the orbiting body having a Z component in the direction of the north ecliptic pole

ω : argument of the perihelion; the angle, subtended at the sun, measured in the plane of the orbit and in the direction of motion, from the ascending node to the perihelion of the orbit

T : the time of perihelion passage, typically presented in Julian Date format, but can be translated into a decimal calendar date

As examples, here are the elements of Earth's orbit, for epoch JD 2458792.5 (or midnight beginning 5 November 2019)

a = 0.9999951820728348 AU
e = 0.01674899215492258
i = 0.02633205404161869°
Ω = 176.9917546445248°
ω = 286.0839149800637°
T = 2458852.774528838694 = 6h 35m 19s on 4 Jan 2020 UTC

and here are the elements of 2I/Borisov orbit, for epoch JD 2458792.5

a = −0.8513198164554499
e = 3.357068272255771
i = 44.05161909545966°
Ω = 308.1483096529710°
ω = 209.1213073058442°
T = 2458826.048866978846 = 13h 10m 22s on 8 Dec 2019 UTC

FINDING THE POSITION AND VELOCITY FROM THE ORBITAL ELEMENTS AND A SPECIFIED TIME...

The solar gravitational parameter,
GM = 1.32712440018e+20 m³ sec⁻²

The conversion factor from astronomical units to meters,
U = 1.495978707e+11 meters/AU

Let the specified time, t, be 8h 52m on 11 December 2019, or
t = JD 2458828.86944

For ELLIPTICAL orbits...

Calculate the period of the orbit, P, in days
P = (π/43200) √[(aU)³/GM]

Calculate the mean anomaly in the orbit at time t
M = 2π(t−T)/P
if M<0 then M=M+2π

Obtain an initial approximation for the eccentric anomaly, u

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

Refine the eccentric anomaly to the exact value

Repeat
U = u
F₁ = u − e sin(u) − M
F₂ = 1 − e cos(u)
F₃ = e sin(u)
F₄ = e cos(u)
D₁ = −F₁/F₂
D₂ = −F₁/(F₂ + D₁F₃/2)
D₃ = −F₁/(F₂ + D₁F₃/2 + F₄D₂²/6)
u = u + D₃
Until |u−U|<0.0000000001

Find the canonical position of the object in its own orbit

x''' = a[cos(u)−e]
y''' = a sin(u) √(1−e²)

Find the true anomaly, θ, in radians

θ' = arctan(y'''/x''')
if x'''>0 and y'''>0 then θ=θ'
if x'''<0 then θ=θ'+π
if x'''>0 and y'''<0 then θ=θ'+2π

Rotate the canonical position vector into heliocentric ecliptic coordinates

x'' = x''' cos(ω) − y''' sin(ω)
y'' = x''' sin(ω) + y''' cos(ω)
x' = x''
y' = y'' cos(i)
z' = z'' sin(i)
x = x' cos(Ω) − y' sin(Ω)
y = x' sin(Ω) + y' cos(Ω)
z = z'

Find the heliocentric distance, r

r = √(x² + y² + z²)

Find the heliocentric ecliptic longitude, λ

λ' = arctan(y/x)
if x>0 and y>0 then λ=λ'
if x<0 then λ=λ+π
if x>0 and y<0 then λ=λ+2π

Find the heliocentric ecliptic latitude, β

β = arcsin(z/r)

Find the canonical velocity of the object in its own orbit

K = √{GM/[aU(1−e²)]}
Vx''' = −K sin(θ)
Vy''' = +K [e + cos(θ)]

Rotate the canonical velocity vector into heliocentric ecliptic coordinates

Vx'' = Vx''' cos(ω) − Vy''' sin(ω)
Vy'' = Vx''' sin(ω) + Vy''' cos(ω)
Vx' = Vx''
Vy' = Vy'' cos(i)
Vz' = Vz'' sin(i)
Vx = Vx' cos(Ω) − Vy' sin(Ω)
Vy = Vx' sin(Ω) + Vy' cos(Ω)
Vz = Vz'

Find the sun-relative speed, V

V = √(Vx² + Vy² + Vz²)

(elliptical orbit treatment ends here)

For HYPERBOLIC orbits...

Calculate the hyperbolic mean anomaly of the orbit,
M = 0.01720209895 (t−T) √[1/(−a)³]

Initialize the variable for the eccentric anomaly, u, as zero.

u = 0
U = 9

Refine the eccentric anomaly to the exact value

While |u−U|>0.0000000001 do
U = u
F₁ = e sinh(u) − u − M
F₂ = e cosh(u) − 1
F₃ = e sinh(u)
F₄ = e cosh(u)
D₁ = −F₁/F₂
D₂ = −F₁/(F₂ + D₁F₃/2)
D₃ = −F₁/(F₂ + D₁F₃/2 + F₄D₂²/6)
u = u + D₃
end while

Find the true anomaly, θ

if M≥0 then
θ = arccos{[e−cosh(u)]/[e cosh(u)−1]}
else
θ = −arccos{[e−cosh(u)]/[e cosh(u)−1]}

Find the heliocentric distance, r

r = a[1−e cosh(u)]

Calculate the canonical position of the object in its own orbit

x''' = r cos θ
y''' = r sin θ

Rotate the canonical position vector into heliocentric ecliptic coordinates

x'' = x''' cos(ω) − y''' sin(ω)
y'' = x''' sin(ω) + y''' cos(ω)
x' = x''
y' = y'' cos(i)
z' = z'' sin(i)
x = x' cos(Ω) − y' sin(Ω)
y = x' sin(Ω) + y' cos(Ω)
z = z'

Find the heliocentric distance, r

r = √(x² + y² + z²)

Find the heliocentric ecliptic longitude, λ

λ' = arctan(y/x)
if x>0 and y>0 then λ=λ'
if x<0 then λ=λ+π
if x>0 and y<0 then λ=λ+2π

Find the heliocentric ecliptic latitude, β

β = arcsin(z/r)

Find the canonical velocity of the object in its own orbit

K = √[GM/(−aU)]
Vx''' = (a/r) K sinh(u)
Vy''' = −(a/r) K √(e²−1) cosh(u)

Rotate the canonical velocity vector into heliocentric ecliptic coordinates

Vx'' = Vx''' cos(ω) − Vy''' sin(ω)
Vy'' = Vx''' sin(ω) + Vy''' cos(ω)
Vx' = Vx''
Vy' = Vy'' cos(i)
Vz' = Vz'' sin(i)
Vx = Vx' cos(Ω) − Vy' sin(Ω)
Vy = Vx' sin(Ω) + Vy' cos(Ω)
Vz = Vz'

Find the sun-relative speed, V

V = √(Vx² + Vy² + Vz²)

(hyperbolic orbit treatment ends here)
 

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