Useful physics equations for military system analysis

Speculation on military systems often turns into an exchange of uninformed guesstimates, obtained by eyeballing measurements. While in most cases it is the best that can be done without access to classified information, physical laws still apply to those systems. A handful of physics formula help constrain their performance, so I have gathered a few here (mostly for my future use). They deal with radar and optical sensors, ballistics and finally aerodynamics.

 

Radar

Radar horizon

img714.gif

2018-05-26 13_59_16-Document1 - Microsoft Word

R is the radar horizon range in kilometers, h is the antenna height in meter, and a is the target altitude in meter. It limits the detection range of radars (and optical sensors too) no matter their power.

 

Radar equation

2018-05-26 21_39_17-radar equation - Recherche Google

with R the maximum range a target of radar cross-section σ can be detected by a radar with minimal detectable power Pe. Ps is the emitted power, G the antenna’s gain,  λ is the wavelength of the radar. A more complete form of the equation gives the radar’s signal to noise ratio (SNR) for the same target:

2018-05-26 21_41_31-Correct Radar Range Equation for SNR at Detection - Signal Processing Stack Exch

With Pn being the power of the electronic noise, Pt the emitted power and Ps the power of the target’s return, Gt and Gr the gains on transmission and reception (assumed to be the same in the first formula), k the Boltzmann constant, To the temperature of the receiver or of the background the radar is looking at, B the bandwidth of the radar, Fn the noise figure of the receiver and L the emission+reception losses between the antenna and the electronics.

 

Antenna gain

2018-05-26 21_52_24-Antenna aperture - Wikipedia.png

G is the gain of an antenna of effective area Aeff (or of physical area Aphys and efficiency Ea) at wavelength λ. It is limited by the diffraction of radio waves.

 

Radar cross section of a flat surface

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Radar cross section σ of a rectangular reflective plate of dimensions Lx and Ly, seen at the angle (θx,θy) and at wavelength λ. A larger surface has a larger RCS when viewed head-on but the energy is spread over a narrower angle. sinc is the cardinal sine function.

 

Optics

Resolution of a telescope (works for radar too)

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D is the distance to the target, a is the aperture (diameter/width) of the sensor, λ is the wavelength of the sensor. This is the maximum resolution, limited by the diffraction of electromagnetic waves.

 

Minimum size of a focused beam

2018-05-26 21_32_19-Document1 - Microsoft Word

2018-05-26-21_32_19-document1-microsoft-word1-e1527363258907.png is the full width of the spot beam of a laser of wavelength λ focusing at a distance f using a mirror of diameter d. The spot can be much smaller than the mirror, so the laser can concentrate its energy and destroy materials much tougher than the mirror’s material.

 

 

Ballistics

Speed of a rocket after burnout (Tsiolkovski’s equation)

2018-05-26 13_54_06-Tsiolkovsky rocket equation - Wikipedia.png

Δ is the speed difference between before ignition and at burnout, v_eis the ejection speed of the rocket’s gasses (usually around 2000m/s for solid rockets, and usually reported as the specific impulse), m_{i}is the initial mass of the rocket (fuel + the rest) before ignition, and m_f the final mass after burnout

 

Range of a rocket

2018-05-26 13_54_48-Portée (balistique) — Wikipédia

d is the distance crossed by a rocket in parabolic flight, ignoring aerodynamic forces. v is the initial speed of the rocket, g is the acceleration of gravity and θ the firing angle (0 for horizontal firing). It is only valid for ranges far smaller than the perimeter of the Earth.

 

Time of flight

2018-05-26 13_54_40-Portée (balistique) — Wikipédia

t is the duration the rocket spend in flight, the rest is the same as above.

 

Max range

2018-05-26 13_54_48-Portée (balistique) — Wikipédia

This is the maximum range of a rocket, obtained when θ = 45°, and is valid for ranges far smaller than the perimeter of the Earth. For long ranges, the relationship is this:

2018-11-01 20_20_26-Window

and the injection angle needed to achieve it is given by

2018-11-01 20_20_03-Window.png

Max altitude at max range

2018-05-26 14_06_51-Document1 - Microsoft Word

For the maximum range trajectory, the rocket reaches a maximum altitude of h

 

Max altitude with a purely vertical trajectory

2018-05-26 14_10_40-Parabole de sûreté — Wikipédia

Vo is the initial speed, and h is the maximum altitude reached when firing with θ = 90°

 

Aerodynamics

Range of a jet aircraft (Breguet’s range equation)

img1614

V is the speed of the aircraft, L/D is lift/drag coefficient, Isp the specific impulse of its engines, Wi the mass at take-off and Wf the mass at landing

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