#### Dynamics of Homogeneous Fleet Combat:

- In EVE, as in life, the only thing you can't go get more of is time. Analyzing fleet combat from the perspective of rates and time
- For homogeneous fleets in combat, the rates of target destruction are determined by the ratio of each fleet's figures of merit, or FOM: $FOM = N \times \sqrt{ DPS \times EHP}$. Some notes on the FOM concept:
- The number of ships in the fleet has squared benefit over ship quality, i.e. $DPS \times HP$ product. This makes obvious intuitive sense considering that bringing one more ship means you have also brought more $EHP$ and more $DPS$, both.
- Ship quality factors, i.e. effective hit points and effective damage per second, are interchangeable in this dynamic model. In reality, of course, they are not because our definition for $T_{kill}$ in this model assumes that the logistics repair rate is constant.
- It is usually impractical to calculate a fleet figure of merit during combat. Instead, by comparing the fractional rate of loss for your own fleet and your opponent's fleet for any time period, you are comparing the effective fleet figures of merit. Methods along these lines are already widely used.
- With a model for fleet advantage, it is possible to quantify the risk aversion of fleet commanders by comparing their fleet FOM against hostile fleets FOM where they choose whether or not to engage.
- When fighting opponents in same-size or smaller-size targets, prelocking secondaries can have a beneficial effect by reducing total kill time.

#### Ship Bumping and Mass-Matching Approaches [November 2015, January 2016]

- Ship bumping in EVE is based on elastic collisions. Data confirms this at low energies (adiabatic case).
- Using a mass-matching intermediate ship to transform the bump energy to the target mass improves bump energy transfer and total bump distance.
- For one intermediate ship, the optimal intermediate mass is the geometric mean of the bumping and target ships, $\large m_{2,optimal} = \sqrt{m_1 m_3}$
- At higher velocities, a double-bump phenomenon is occurring, delivering multiples of the input energy to the target (nonadiabatic case).
- Theory for N-ship adiabatic case is also presented and a solution suggested.

#### Refinement to Fundamental Orbit Limits [Nov 2015]

- Orbits are limited either by the maximum velocity of the ship, or by the maximum acceleration that it can achieve.
- Wide orbits, where the orbit radius is much larger than the characteristic distance of the ship,
*τ*, are limited by maximum ship velocity, $\large \omega \leq V_{MAX} / R$ .*V*_{MAX} - Close-in orbits, where the orbit radius is much closer than the characteristic distance of the ship,
*τ*, are limited by maximum ship acceleration, $\large \omega \leq \sqrt{\frac{V_{MAX}}{\tau R} }$ .*V*_{MAX}

#### Part I: Ship Motion Basics [March 2010, September 2015]

- When a command vector is entered into the eve interface, by double-clicking in space, aligning or approaching a stationary object, the ship accelerates in the direction of the command vector.
- A ship's maximum acceleration is $\large \frac{V_{MAX}}{\tau}$ or $\large \frac{V_{MAX}}{M \, I}$. Afterburners and micro-warp drives add mass to your ship, dynamically changing the maneuvering time-constant.
- A drag force acts in the opposite direction to the ship's velocity. The magnitude of the drag force is proportional to the ship's velocity and a coefficient. The drag coefficient is the reciprocal of the ships 'inertia factor', $\large \sigma = 1/I$, and has units $\large \left [ \frac{10^6 kg}{s} \right ]$. This 'inertia factor' term is affected by several different skills.
- As the ship's velocity increases, the drag force increases until the ship reaches a terminal velocity,
*V*. Base velocity is affected usually by one or more skills._{MAX} - Acceleration and deceleration exhibit symmetric dynamics (i.e. they have the same time-constant). Unlike driving a car, the mechanism for both acceleration and deceleration motion in EVE are the same.
- The time-to-warp is a linear multiple of the time-constant,
*τ*. Simply put,*τ*is the time for velocity to reach*1 - e*, or about 62.3% of^{-1}*V*. Time-to-warp is the time required to get to 75% of_{MAX}*V*._{MAX} - Base Cruising Speed is not affected by mass unless afterburners or micro-warp drives are active. This breaks somewhat with the force-mass linear drag model used for ship movement. MWD/AB Speed is dependent on mass of the ship and module as well as the effect of armor plates.
- The stopping distance of a ship moving a a velocity
*V*, is just the product of the maneuvering time constant,*τ*, and*V*, or*τ*x*V*.

#### Part II: Analyzing Ship Maneuvers [March 2010, September 2015]

- 'Orbit at R' accelerates towards a point R away from the target object. Orbit behavior is well described by the linear drag equations and a centripetal force balance model.
- Tighter orbits can be achieved by manually piloting.

#### Part III: Ship Interactions and Bumping [March 2010, September 2015]

- The effect of 'Approach' and 'Keep At Range' commands on distance to target and angular velocity are well modeled for orbiting attackers.
- Collisions in EVE are elastic, meaning energy is conserved. Velocity of the bumped ship, immediately after the collision is determined by, $\large v(t=0^+) = \frac{2 v_1 m_1}{m_1 + m_2}$.
- Successful bumpers should
~~maximize velocity and mass for best results~~maximize their velocity given that they have the same mass as the bump target. If a bumping ship is widely mismatched in mass from the bump target, it is best to use an intermediate ship to match the mass with the bump target. - Webbing of a ship that is being bumped does not affect the distance that it is bumped; however, it will affect it’s ability to accelerate itself.

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