diff --git a/DIRECTORY.md b/DIRECTORY.md index 133a1ab01..29514579c 100644 --- a/DIRECTORY.md +++ b/DIRECTORY.md @@ -741,6 +741,7 @@ ## 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) diff --git a/physics/basic_orbital_capture.py b/physics/basic_orbital_capture.py new file mode 100644 index 000000000..eeb45e602 --- /dev/null +++ b/physics/basic_orbital_capture.py @@ -0,0 +1,178 @@ +from math import pow, sqrt + +from scipy.constants import G, c, pi + +""" +These two functions will return the radii of impact 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 impact 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: + """ + Input Params: + ------------- + target_body_radius: Radius of the central body SI units: meters | m + target_body_mass: Mass of the central body SI units: kilograms | kg + projectile_velocity: Velocity of object moving toward central body + SI units: meters/second | m/s + Returns: + -------- + >>> capture_radii(6.957e8, 1.99e30, 25000.0) + 17209590691.0 + >>> capture_radii(-6.957e8, 1.99e30, 25000.0) + Traceback (most recent call last): + ... + ValueError: Radius cannot be less than 0 + >>> capture_radii(6.957e8, -1.99e30, 25000.0) + Traceback (most recent call last): + ... + ValueError: Mass cannot be less than 0 + >>> capture_radii(6.957e8, 1.99e30, c+1) + Traceback (most recent call last): + ... + ValueError: Cannot go beyond speed of light + + Returned SI units: + ------------------ + meters | m + """ + + if target_body_mass < 0: + raise ValueError("Mass cannot be less than 0") + if target_body_radius < 0: + raise ValueError("Radius cannot be less than 0") + if projectile_velocity > c: + raise ValueError("Cannot go beyond speed of light") + + escape_velocity_squared = (2 * G * target_body_mass) / target_body_radius + capture_radius = target_body_radius * sqrt( + 1 + escape_velocity_squared / pow(projectile_velocity, 2) + ) + return round(capture_radius, 0) + + +def capture_area(capture_radius: float) -> float: + """ + Input Param: + ------------ + capture_radius: The radius of orbital capture and impact for a central body of + mass M and a projectile moving towards it with velocity v + SI units: meters | m + Returns: + -------- + >>> capture_area(17209590691) + 9.304455331329126e+20 + >>> capture_area(-1) + Traceback (most recent call last): + ... + ValueError: Cannot have a capture radius less than 0 + + Returned SI units: + ------------------ + meters*meters | m**2 + """ + + if capture_radius < 0: + raise ValueError("Cannot have a capture radius less than 0") + sigma = pi * pow(capture_radius, 2) + return round(sigma, 0) + + +if __name__ == "__main__": + from doctest import testmod + + testmod() + +""" +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 impact, 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 +"""