Physics/basic orbital capture

DIRECTORY.md

@@ -745,6 +745,7 @@
 
 ## Physics
   * [Archimedes Principle](physics/archimedes_principle.py)
+  * [Basic Orbital Capture](physics/basic_orbital_capture.py)
   * [Casimir Effect](physics/casimir_effect.py)
   * [Centripetal Force](physics/centripetal_force.py)
   * [Grahams Law](physics/grahams_law.py)

physics/basic_orbital_capture.py

@@ -0,0 +1,123 @@
+import math
+
+"""
+These two functions will return the radii of capture for a target object
+of mass M and radius R as well as it's effective cross sectional area σ(sigma).
+That is to say any projectile with velocity v passing within σ, will be caputered
+by the target object with mass M. The derivation of which is given at the bottom
+of this file.
+
+The derivation shows that a projectile does not need to aim directly at the target
+body in order to hit it, as  R_capture>R_target. Astronomers refer to the effective
+cross section for capture as σ=π*R_capture**2.
+
+This algorithm does not account for an N-body problem.
+
+"""
+
+
+def capture_radii(
+    target_body_radius: float, target_body_mass: float, projectile_velocity: float
+) -> float:
+    # Gravitational constant to four significant figures as of 7/8/2023|
+    # Source google: gravitational constant
+    g = 6.6743e-11  # SI units (N*m**2)/kg**2
+
+<<<<<<< HEAD
+    #Gravitational constant to four significant figures as of 7/8/2023|
+    #Source google: gravitational constant
+    g=6.6743e-11    #SI units (N*m**2)/kg**2
+=======
+    escape_velocity_squared = (2 * g * target_body_mass) / target_body_radius
+
+    capture_radius = target_body_radius * math.sqrt(
+        1 + escape_velocity_squared / math.pow(projectile_velocity, 2)
+    )
+    return capture_radius
+
+
+def capture_area(capture_radius: float) -> float:
+    sigma = math.pi * math.pow(capture_radius, 2)
+    return sigma
+
+
+"""
+Derivation:
+
+Let: Mt=target mass, Rt=target radius, v=projectile_velocity,
+     r_0=radius of projectile at instant 0 to CM of target
+     v_p=v at closest approach,
+     r_p=radius from projectile to target CM at closest approach,
+     R_capture= radius of impact for projectile with velocity v
+
+(1)At time=0  the projectile's energy falling from infinity| E=K+U=0.5*m*(v**2)+0
+
+    E_initial=0.5*m*(v**2)
+
+(2)at time=0 the angular momentum of the projectile relative to CM target|
+    L_initial=m*r_0*v*sin(Θ)->m*r_0*v*(R_capture/r_0)->m*v*R_capture
+
+    L_i=m*v*R_capture
+
+(3)The energy of the projectile at closest approach will be its kinetic energy
+   at closest approach plus gravitational potential energy(-(GMm)/R)|
+    E_p=K_p+U_p->E_p=0.5*m*(v_p**2)-(G*Mt*m)/r_p
+
+    E_p=0.0.5*m*(v_p**2)-(G*Mt*m)/r_p
+
+(4)The angular momentum of the projectile relative to the target at closest
+   approach will be L_p=m*r_p*v_p*sin(Θ), however relative to the target Θ=90°
+   sin(90°)=1|
+
+    L_p=m*r_p*v_p
+(5)Using conservation of angular momentum and energy, we can write a quadratic
+   equation that solves for r_p|
+
+   (a)
+    Ei=Ep-> 0.5*m*(v**2)=0.5*m*(v_p**2)-(G*Mt*m)/r_p-> v**2=v_p**2-(2*G*Mt)/r_p
+
+   (b)
+    Li=Lp-> m*v*R_capture=m*r_p*v_p-> v*R_capture=r_p*v_p-> v_p=(v*R_capture)/r_p
+
+   (c) b plugs int a|
+    v**2=((v*R_capture)/r_p)**2-(2*G*Mt)/r_p->
+
+    v**2-(v**2)*(R_c**2)/(r_p**2)+(2*G*Mt)/r_p=0->
+
+    (v**2)*(r_p**2)+2*G*Mt*r_p-(v**2)*(R_c**2)=0
+
+   (d) Using the quadratic formula, we'll solve for r_p then rearrange to solve to
+       R_capture
+
+    r_p=(-2*G*Mt ± sqrt(4*G^2*Mt^2+ 4(v^4*R_c^2)))/(2*v^2)->
+
+    r_p=(-G*Mt ± sqrt(G^2*Mt+v^4*R_c^2))/v^2->
+
+    r_p<0 is something we can ignore, as it has no physical meaning for our purposes.->
+
+    r_p=(-G*Mt)/v^2 + sqrt(G^2*Mt^2/v^4 + R_c^2)
+
+   (e)We are trying to solve for R_c. We are looking for capture, so we want r_p=Rt
+
+    Rt + G*Mt/v^2 = sqrt(G^2*Mt^2/v^4 + R_c^2)->
+
+    (Rt + G*Mt/v^2)^2 = G^2*Mt^2/v^4 + R_c^2->
+
+    Rt^2 + 2*G*Mt*Rt/v^2 + G^2*Mt^2/v^4 = G^2*Mt^2/v^4 + R_c^2->
+
+    Rt**2 + 2*G*Mt*Rt/v**2 = R_c**2->
+
+    Rt**2 * (1 + 2*G*Mt/Rt *1/v**2) = R_c**2->
+
+    escape velocity = sqrt(2GM/R)= v_escape**2=2GM/R->
+
+    Rt**2 * (1 + v_esc**2/v**2) = R_c**2->
+
+(6)
+    R_capture = Rt * sqrt(1 + v_esc**2/v**2)
+
+Source: Problem Set 3 #8 c.Fall_2017|Honors Astronomy|Professor Rachel Bezanson
+
+Source #2: http://www.nssc.ac.cn/wxzygx/weixin/201607/P020160718380095698873.pdf
+           8.8 Planetary Rendezvous: Pg.368
+"""