# Difference between distance from Mars (Rmag) and B-plane R coordinate

I am running some GMAT simulations on Mars 2020 trajectory to Mars, following the ephemerides published by JPL. I create a target sequence and target the planet using the b-plane coordinates B and T. I noticed that when setting T=0 and B=24000km (or any other value, for that matter), the spacecraft ends up at a distance from the centre of Mars that is smaller than this value, about 4000km.

The GMAT help file doesn't go into detail about the B parameters but mentions that w x r effects of rotating systems might not be simulated in some implementations. I tried changing the martian coordinate system from MJ2000Eq to ICRF and the results are identical. I don't understand how this difference is created or how to configure the goal commands so that I can target a specific distance from the centre reliably, using the B-plane? Rmag can be used in place of B or T, but it doesn't convey the side of the planet in which the spacecraft will be inserted

A good overview of what the B-plane is and how it's defined is available from the FreeFlyer guide.

The "B" in B-plane is the hyperbolic impact parameter, the semi-minor axis of the hyperbola that is the spacecraft's orbit. Importantly, as the equations from Wikipedia show, the impact parameter alone is not enough to determine the periapsis of the orbit. The other parameter required in the hyperbolic excess velocity, $$v_{\infty}$$, which is also the square root of the characteristic energy, $$C3$$.

You can think of the impact parameter as "the distance by which a body, if it continued on an unperturbed path, would miss the central body at its closest approach." (Wikipedia)

Unperturbed meaning we ignore the planet's (Mars) gravity in the B-plane representation. Mars' gravity will pull the spacecraft closer and therefore the close approach distance, $$r_p$$, is always less than the impact parameter, $$b$$, value.

Wikipedia gives a neat equation relating $$r_p$$ and $$b$$: $$r_p=\frac{\mu}{v_{\infty}^2} \cdot (\sqrt{1+(\frac{b \cdot v_{\infty}^2}{\mu})^2}-1)$$ Which could be rearranged for $$b$$: $$b=\frac{\mu}{v_{\infty}^2} \cdot \sqrt{(1+\frac{r_p \cdot v_{\infty}^2}{\mu})^2-1}$$

In GMAT I started with a Mars relative state vector for Mars 2020 from HORIZONS on February 1st:

            JDUT ,            Calendar Date (UT ),                      X,                      Y,                      Z,                     VX,                     VY,                     VZ,
**************************************************************************************************************************************************************************************************
$$SOE 2459246.500000000, A.D. 2021-Feb-01 00:00:00.0000, -3.270678971433882E+06, -1.333631118481763E+06, 1.800947903043214E+06, 2.089538719640068E+00, 8.723492698345683E-01, -1.146618806007647E+00,$$EOE
**************************************************************************************************************************************************************************************************


I'll assume you can create the necessary GMAT components (spacecraft, propagators, coordinate frames, etc.) and just give you the mission sequence code:

%----------------------------------------
%---------- Arrays, Variables, Strings
%----------------------------------------

Create Variable C3 mu rp B dV angle BT BR;
GMAT mu = 42828.37; % GM for Mars
GMAT rp = 3389.5; % Mars target periapsis (km)

%----------------------------------------
%---------- Mission Sequence
%----------------------------------------

BeginMissionSequence;

GMAT angle = DegToRad(0); % B-Plane theta angle (radians, input is degrees)

% Target TCM
Target 'Target Mars B-Plane' DC1 {SolveMode = Solve, ExitMode = DiscardAndContinue, ShowProgressWindow = true};
% Vary TCM
Vary 'Vary TCM.V' DC1(TCM.Element1 = 0, {Perturbation = 1e-2, Lower = -0.05, Upper = 0.05, MaxStep = .01, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0});
Vary 'Vary TCM.N' DC1(TCM.Element2 = 0, {Perturbation = 1e-2, Lower = -0.05, Upper = 0.05, MaxStep = .01, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0});
Vary 'Vary TCM.B' DC1(TCM.Element3 = 0, {Perturbation = 1e-2, Lower = -0.05, Upper = 0.05, MaxStep = .01, AdditiveScaleFactor = 0.0, MultiplicativeScaleFactor = 1.0});
Maneuver 'TCM' TCM(M2020);

% Propagate
Propagate 'Prop to periapsis' MarsFull(M2020) {M2020.Mars.Periapsis, StopTolerance = 1e-05};

% Script
BeginScript
GMAT C3 = M2020.Mars.IncomingC3Energy;
GMAT B = mu/C3*sqrt((1+rp*C3/mu)^2-1);
GMAT BT = B*cos(angle);
GMAT BR = -B*sin(angle); % recall that R is positive "down"
GMAT dV = sqrt(TCM.Element1^2+TCM.Element2^2+TCM.Element3^2)*1000; % TCM dV (m/s)
EndScript;

% Achieve Targets
Achieve 'Achieve B.T' DC1(M2020.MarsMJ2000Ec.BdotT = BT, {Tolerance = 1});
Achieve 'Achieve B.R' DC1(M2020.MarsMJ2000Ec.BdotR = BR, {Tolerance = 1});
EndTarget

Write dV { Style = Concise, LogFile = false, MessageWindow = true }


I moved from an explicit "B.T & B.R" targeting to a perhaps more intuitive "B & $$\theta$$" representation. Here are a few example output plots, left an angle of 0°, and right an angle of 135°:

• – uhoh
Feb 16, 2022 at 1:03