void Ephemeris<Frame>::ComputeApsides(not_null<MassiveBody const*> const body1,
not_null<MassiveBody const*> const body2,
DiscreteTrajectory<Frame>& apoapsides1,
DiscreteTrajectory<Frame>& periapsides1,
DiscreteTrajectory<Frame>& apoapsides2,
DiscreteTrajectory<Frame>& periapsides2) {
not_null<ContinuousTrajectory<Frame> const*> const body1_trajectory =
trajectory(body1);
not_null<ContinuousTrajectory<Frame> const*> const body2_trajectory =
trajectory(body2);
typename ContinuousTrajectory<Frame>::Hint hint1;
typename ContinuousTrajectory<Frame>::Hint hint2;
// Computes the derivative of the squared distance between |body1| and |body2|
// at time |t|.
auto const evaluate_square_distance_derivative =
[body1_trajectory, body2_trajectory, &hint1, &hint2](
Instant const& t) -> Variation<Square<Length>> {
DegreesOfFreedom<Frame> const body1_degrees_of_freedom =
body1_trajectory->EvaluateDegreesOfFreedom(t, &hint1);
DegreesOfFreedom<Frame> const body2_degrees_of_freedom =
body2_trajectory->EvaluateDegreesOfFreedom(t, &hint2);
RelativeDegreesOfFreedom<Frame> const relative =
body1_degrees_of_freedom - body2_degrees_of_freedom;
return 2.0 * InnerProduct(relative.displacement(), relative.velocity());
};
std::experimental::optional<Instant> previous_time;
std::experimental::optional<Variation<Square<Length>>>
previous_squared_distance_derivative;
for (Instant time = t_min(); time <= t_max(); time += parameters_.step()) {
Variation<Square<Length>> const squared_distance_derivative =
evaluate_square_distance_derivative(time);
if (previous_squared_distance_derivative &&
Sign(squared_distance_derivative) !=
Sign(*previous_squared_distance_derivative)) {
CHECK(previous_time);
// The derivative of |squared_distance| changed sign. Find its zero by
// bisection, this is the time of the apsis. Then compute the apsis and
// append it to one of the output trajectories.
Instant const apsis_time = Bisect(evaluate_square_distance_derivative,
*previous_time,
time);
DegreesOfFreedom<Frame> const apsis1_degrees_of_freedom =
body1_trajectory->EvaluateDegreesOfFreedom(apsis_time, &hint1);
DegreesOfFreedom<Frame> const apsis2_degrees_of_freedom =
body2_trajectory->EvaluateDegreesOfFreedom(apsis_time, &hint2);
if (Sign(squared_distance_derivative).Negative()) {
apoapsides1.Append(apsis_time, apsis1_degrees_of_freedom);
apoapsides2.Append(apsis_time, apsis2_degrees_of_freedom);
} else {
periapsides1.Append(apsis_time, apsis1_degrees_of_freedom);
periapsides2.Append(apsis_time, apsis2_degrees_of_freedom);
}
}
previous_time = time;
previous_squared_distance_derivative = squared_distance_derivative;
}
}
void Ephemeris<Frame>::ComputeApsides(
not_null<MassiveBody const*> const body,
typename DiscreteTrajectory<Frame>::Iterator const begin,
typename DiscreteTrajectory<Frame>::Iterator const end,
DiscreteTrajectory<Frame>& apoapsides,
DiscreteTrajectory<Frame>& periapsides) {
not_null<ContinuousTrajectory<Frame> const*> const body_trajectory =
trajectory(body);
typename ContinuousTrajectory<Frame>::Hint hint;
std::experimental::optional<Instant> previous_time;
std::experimental::optional<DegreesOfFreedom<Frame>>
previous_degrees_of_freedom;
std::experimental::optional<Square<Length>> previous_squared_distance;
std::experimental::optional<Variation<Square<Length>>>
previous_squared_distance_derivative;
for (auto it = begin; it != end; ++it) {
Instant const time = it.time();
DegreesOfFreedom<Frame> const degrees_of_freedom = it.degrees_of_freedom();
DegreesOfFreedom<Frame> const body_degrees_of_freedom =
body_trajectory->EvaluateDegreesOfFreedom(time, &hint);
RelativeDegreesOfFreedom<Frame> const relative =
degrees_of_freedom - body_degrees_of_freedom;
Square<Length> const squared_distance =
InnerProduct(relative.displacement(), relative.displacement());
// This is the derivative of |squared_distance|.
Variation<Square<Length>> const squared_distance_derivative =
2.0 * InnerProduct(relative.displacement(), relative.velocity());
if (previous_squared_distance_derivative &&
Sign(squared_distance_derivative) !=
Sign(*previous_squared_distance_derivative)) {
CHECK(previous_time &&
previous_degrees_of_freedom &&
previous_squared_distance);
// The derivative of |squared_distance| changed sign. Construct a Hermite
// approximation of |squared_distance| and find its extrema.
Hermite3<Instant, Square<Length>> const
squared_distance_approximation(
{*previous_time, time},
{*previous_squared_distance, squared_distance},
{*previous_squared_distance_derivative,
squared_distance_derivative});
std::set<Instant> const extrema =
squared_distance_approximation.FindExtrema();
// Now look at the extrema and check that exactly one is in the required
// time interval. This is normally the case, but it can fail due to
// ill-conditioning.
Instant apsis_time;
int valid_extrema = 0;
for (auto const& extremum : extrema) {
if (extremum >= *previous_time && extremum <= time) {
apsis_time = extremum;
++valid_extrema;
}
}
if (valid_extrema != 1) {
// Something went wrong when finding the extrema of
// |squared_distance_approximation|. Use a linear interpolation of
// |squared_distance_derivative| instead.
apsis_time = Barycentre<Instant, Variation<Square<Length>>>(
{time, *previous_time},
{*previous_squared_distance_derivative,
-squared_distance_derivative});
}
// Now that we know the time of the apsis, construct a Hermite
// approximation of the position of the body, and use it to derive its
// degrees of freedom. Note that an extremum of
// |squared_distance_approximation| is in general not an extremum for
// |position_approximation|: the distance computed using the latter is a
// 6th-degree polynomial. However, approximating this polynomial using a
// 3rd-degree polynomial would yield |squared_distance_approximation|, so
// we shouldn't be far from the truth.
Hermite3<Instant, Position<Frame>> position_approximation(
{*previous_time, time},
{previous_degrees_of_freedom->position(),
degrees_of_freedom.position()},
{previous_degrees_of_freedom->velocity(),
degrees_of_freedom.velocity()});
DegreesOfFreedom<Frame> const apsis_degrees_of_freedom(
position_approximation.Evaluate(apsis_time),
position_approximation.EvaluateDerivative(apsis_time));
if (Sign(squared_distance_derivative).Negative()) {
apoapsides.Append(apsis_time, apsis_degrees_of_freedom);
} else {
periapsides.Append(apsis_time, apsis_degrees_of_freedom);
}
}
previous_time = time;
previous_degrees_of_freedom = degrees_of_freedom;
previous_squared_distance = squared_distance;
previous_squared_distance_derivative = squared_distance_derivative;
}
}