CAAP: Constant Acceleration Accounted Perspective

• CAAP is a method that defines CME expansion by propagating leading, center, and trailing-edge speeds to a common epoch, enabling instantaneous expansion estimates.
• It employs a constant acceleration assumption for CME substructures to correct biases found in conventional time-independent analyses.
• Validation using Solar Orbiter and Wind data shows CAAP’s (V_C - V_T) estimate of 67 km/s closely matches a directly measured 68 km/s, improving space weather interpretations.

Searching arXiv for the specified paper and closely related CAAP/CME expansion work. CAAP, the Constant Acceleration Accounted Perspective, is a method for estimating the instantaneous expansion speed of a coronal mass ejection (CME) from in situ observations by explicitly accounting for the possibility that CME substructures accelerate during spacecraft traversal. It was introduced to address a limitation of conventional single-point expansion analyses, which usually return a time-independent expansion speed inferred from measurements taken at different times across the CME passage. In the validated formulation, CAAP treats the leading edge, center, and trailing edge as kinematic substructures subject to constant acceleration over the interval of interest, then propagates their speeds to a common epoch so that expansion can be defined from nearly simultaneous quantities rather than from temporally separated samples (Agarwal et al., 10 Apr 2026).

1. Definition and problem setting

CAAP was developed in the context of CME and magnetic cloud (MC) analysis, where the quantity of interest is not merely a global speed gradient across the ejecta, but the instantaneous speed of expansion at a particular moment. The distinction is central because a spacecraft samples the CME sequentially: the leading edge is encountered first, then the center, then the trailing edge. If the CME is accelerating or decelerating during that interval, then the speed difference between front and rear portions does not represent the expansion state at any single time (Agarwal et al., 10 Apr 2026).

The method is motivated by the observation that conventional single-point in situ analysis often assumes that CME substructures move at constant speed during the encounter, that the expansion is effectively time-independent during the passage, and that leading-edge and trailing-edge speeds sampled hours apart can be interpreted as if they were simultaneous. The validated CAAP study argues that these assumptions are often not valid. This matters because the true instantaneous expansion affects the CME’s radial size, magnetic field dilution, arrival timing, and duration of geomagnetic disturbance (Agarwal et al., 10 Apr 2026).

In this framework, CAAP is not a general CME reconstruction scheme; it is specifically a kinematic correction method for recovering the instantaneous expansion state from in situ measurements when different CME substructures are observed at different times. A plausible implication is that CAAP is most useful precisely in cases where non-negligible substructure acceleration would bias conventional expansion estimates.

2. Kinematic formulation

CAAP assumes that each CME substructure—leading edge $L$, center $C$, and trailing edge $T$—experiences constant acceleration over the interval of interest. The governing relation is

$V_F(t_j) = V_F(t_i) + a_F\, t_{ji},$

where $F \in \{L,C,T\}$, $a_F$ is the constant acceleration of that substructure, and $t_{ji}=t_j-t_i$ (Agarwal et al., 10 Apr 2026).

The acceleration is estimated from two-spacecraft measurements as

$a_F = \frac{(V_F)_{\text{Wind} - (V_F)_{\text{SolO}}}{(t_F)_{\text{Wind} - (t_F)_{\text{SolO}}}.$

Once $a_F$ is obtained, the measured speed of each substructure can be propagated to a common time, enabling evaluation of the instantaneous expansion from simultaneous substructure speeds rather than from temporally offset observations (Agarwal et al., 10 Apr 2026).

The validated study evaluates three equivalent combinations for instantaneous expansion speed:

$$V_L - V_C,\qquad V_C - V_T,\qquad \frac{V_L - V_T}{2}.$$

For the event analyzed, the physically preferred estimate was the difference between the center and trailing edge because it agreed best with the direct instantaneous measurement (Agarwal et al., 10 Apr 2026). This does not establish that $C$0 is universally optimal; rather, it identifies that combination as the most accurate one for the specific geometry and event examined.

3. Relation to conventional expansion analysis

The principal conceptual distinction between CAAP and conventional methods is the difference between time-dependent instantaneous expansion speed and time-independent expansion speed. Conventional methods often assign a single expansion speed to the whole MC crossing, commonly based on a linear speed profile or on a front–rear speed difference. In the terminology of the validation study, this is a time-independent expansion speed (Agarwal et al., 10 Apr 2026).

CAAP instead reconstructs the expansion at a specific phase of the encounter by transporting substructure velocities to a common epoch using the estimated accelerations. This is particularly relevant when the CME speed profile is non-linear. If the trailing edge accelerates or the leading edge decelerates during traversal, a conventional estimate may be biased high or low because it conflates temporal evolution with spatial structure (Agarwal et al., 10 Apr 2026).

The paper also contrasts CAAP with the method of Demoulin et al. (2020), which models the CME speed profile as a combination of center motion and expansion but assumes no bulk acceleration of the center. CAAP differs in allowing constant acceleration of the substructures themselves (Agarwal et al., 10 Apr 2026). This suggests that CAAP is intended for regimes in which bulk acceleration is not negligible over the spacecraft crossing time.

4. Empirical validation using Solar Orbiter and Wind

The principal validation of CAAP used a CME observed during 3–5 November 2021 by Solar Orbiter (SolO) at 0.85 AU and Wind at 0.98 AU (Agarwal et al., 10 Apr 2026). The spacecraft were separated by 0.13 AU radially and 2.3° angularly, with about 1° in longitude and about 2.1° in latitude. This radial alignment enabled simultaneous sampling of different CME substructures at nearly the same heliolongitude, which is unusual and was crucial for validation (Agarwal et al., 10 Apr 2026).

The rare observational configuration was that Wind observed the MC center while Solar Orbiter observed the MC trailing edge. Because these measurements were nearly simultaneous, they permitted a direct estimate of the instantaneous expansion speed independent of CAAP. The direct measurement was

$C$1

with the timing difference between the two measurements being about 19 minutes (Agarwal et al., 10 Apr 2026).

CAAP was then applied independently to the single-spacecraft SolO and Wind data. The substructure speeds used in the CAAP evaluation were reported as follows (Agarwal et al., 10 Apr 2026):

Spacecraft	Substructure	Speeds
SolO	LE	661, 718, 780 km s$C$2
SolO	Center	594, 616, 640 km s$C$3
SolO	TE	522, 546, 573 km s$C$4
Wind	LE	708, 782, 860 km s$C$5
Wind	Center	612, 641, 671 km s$C$6
Wind	TE	542, 574, 608 km s$C$7

Using CAAP propagation, the inferred instantaneous expansion speeds were (Agarwal et al., 10 Apr 2026):

Expression	SolO	Wind
$C$8	140 km s$C$9	141 km s$T$0
$T$1	67 km s$T$2	67 km s$T$3
$T$4	103 km s$T$5	104 km s$T$6

The central validation result was that the $T$7 CAAP estimate, 67 km s$T$8, matched the directly measured 68 km s$T$9 extremely well (Agarwal et al., 10 Apr 2026). The other two expressions were inconsistent because they depended on the leading-edge speed, which the paper argues may be particularly sensitive to drag and may not be well represented by simple constant-acceleration extrapolation over the whole interval.

5. Quantitative comparison with conventional methods

The same event permits a direct comparison between CAAP and conventional expansion analysis. Using the conventional method, the time-independent expansion speeds were (Agarwal et al., 10 Apr 2026):

• SolO: 44 km s$V_F(t_j) = V_F(t_i) + a_F\, t_{ji},$0
• Wind: 50 km s$V_F(t_j) = V_F(t_i) + a_F\, t_{ji},$1

These values are smaller than both the directly measured instantaneous expansion speed of 68 km s$V_F(t_j) = V_F(t_i) + a_F\, t_{ji},$2 and the CAAP-derived 67 km s$V_F(t_j) = V_F(t_i) + a_F\, t_{ji},$3 estimate from $V_F(t_j) = V_F(t_i) + a_F\, t_{ji},$4 (Agarwal et al., 10 Apr 2026). The paper states that the conventional method underestimates the instantaneous expansion by roughly 15–35 km s$V_F(t_j) = V_F(t_i) + a_F\, t_{ji},$5.

The study further notes a discrepancy with Regnault et al. (2024), who reported 73 km s$V_F(t_j) = V_F(t_i) + a_F\, t_{ji},$6, attributing the difference to the use of the time center in that work, whereas the validated CAAP study uses the size center (Agarwal et al., 10 Apr 2026). This detail is methodologically significant because it shows that even within expansion analyses, the operational definition of “center” can materially affect inferred speeds.

The broader significance is that underestimating instantaneous expansion can lead to underestimation of CME radial size and mischaracterization of geoeffectiveness (Agarwal et al., 10 Apr 2026). This suggests that CAAP is not simply a refinement in notation; it modifies physically consequential quantities used in space-weather interpretation.

6. Role in interpreting CME substructure evolution

The validation paper does not treat CAAP in isolation. It uses the same SolO–Wind conjunction to investigate the temporal evolution of the shock, sheath, and magnetic cloud, and this broader analysis contextualizes why instantaneous expansion speed matters (Agarwal et al., 10 Apr 2026).

The observed timing changes between SolO and Wind were (Agarwal et al., 10 Apr 2026):

• Shock arrival: delay 5.51 hr
• MC leading edge: delay 4.97 hr
• MC center: delay 6.81 hr
• MC trailing edge: delay 8.6 hr

The radial sizes were reported as (Agarwal et al., 10 Apr 2026):

• MC: 39.3 $V_F(t_j) = V_F(t_i) + a_F\, t_{ji},$7 at SolO, 53.5 $V_F(t_j) = V_F(t_i) + a_F\, t_{ji},$8 at Wind
• Sheath: 61.9 $V_F(t_j) = V_F(t_i) + a_F\, t_{ji},$9 at SolO, 61.3 $F \in \{L,C,T\}$0 at Wind

Thus the MC expanded by about 14.2 $F \in \{L,C,T\}$1, whereas the sheath size remained nearly constant (Agarwal et al., 10 Apr 2026). The paper interprets the MC growth as continued expansion between SolO and Wind and notes that all observed arrival times of the LE, center, and TE at Wind were earlier than would be expected under constant-speed propagation, reinforcing the inference that the substructures were accelerating.

In the same event, the shock became unexpectedly stronger at Wind, with reported parameters including shock speed 515 km s$F \in \{L,C,T\}$2 at SolO and 760 km s$F \in \{L,C,T\}$3 at Wind, and more pronounced compression and heating ratios at Wind (Agarwal et al., 10 Apr 2026). The paper argues that this likely reflects a combination of CME-driver acceleration, reduced upstream Alfvén speed at Wind, altered shock geometry, and possibly interaction with a high-speed solar wind stream behind the CME.

These observations matter for CAAP because they show that substructure evolution can be substantial even over 0.13 AU (Agarwal et al., 10 Apr 2026). A plausible implication is that a method assuming temporal stationarity across the crossing is particularly vulnerable in such cases.

7. Magnetic-cloud evolution, orientation, and implications for space weather

The same study links CAAP to magnetic-cloud evolution and to broader space-weather diagnostics. For the MC, the duration changed from 12.5 hr at SolO to 16.1 hr at Wind, and the radial size from 39.3 $F \in \{L,C,T\}$4 to 53.5 $F \in \{L,C,T\}$5, making the MC about 1.4 times larger at Wind (Agarwal et al., 10 Apr 2026). The paper associates this with continued expansion and with a high-speed solar wind stream (HSSS) behind the MC at SolO that appears to have compressed the MC locally and may have helped accelerate it during propagation.

The field and size evolution were analyzed using the power-law relations

$F \in \{L,C,T\}$6

Using $F \in \{L,C,T\}$7 nT and $F \in \{L,C,T\}$8 at SolO, and $F \in \{L,C,T\}$9 nT and $a_F$0 at Wind, the paper derives $a_F$1 and $a_F$2 (Agarwal et al., 10 Apr 2026). Because $a_F$3 exceeds $a_F$4, the radius expanded faster than the field weakened, implying that the MC’s magnetic flux was not conserved. Under a force-free flux-rope assumption, the paper estimates that poloidal magnetic flux increased by about 24% from SolO to Wind, and notes that a previous study found 13% increase in axial flux (Agarwal et al., 10 Apr 2026). The study interprets this as evidence of additional flux injection, likely via magnetic reconnection near the trailing edge.

The paper also reports a discrepancy between visual inspection, which suggested an ENW-type flux rope and hence a highly inclined structure, and minimum variance analysis (MVA), which yielded a low-inclination axis. The reported MVA results were (Agarwal et al., 10 Apr 2026):

• SolO: eigenvalues $a_F$5, $a_F$6, axis orientation $a_F$7
• Wind: eigenvalues $a_F$8, $a_F$9, axis orientation $t_{ji}=t_j-t_i$0

The paper concludes that the MVA-derived axis may be unreliable if the spacecraft crossed the flux rope away from its central axis or if the flux rope was distorted or displaced out of the measurement plane (Agarwal et al., 10 Apr 2026).

For space weather, the validated implications are explicit: instantaneous expansion speed matters, because underestimation affects predictions of CME arrival time, disturbance duration, magnetic flux content, and geoeffectiveness. The study further emphasizes that pre-L1 observations may not fully represent the CME at Earth because shock, sheath, and MC properties can evolve significantly even over a short radial interval (Agarwal et al., 10 Apr 2026).

In that sense, CAAP is best understood as a physically motivated correction to single-spacecraft CME expansion analysis, validated by an unusually favorable multipoint conjunction. Its main demonstrated result is that, for the 3–5 November 2021 event, a CAAP-derived single-spacecraft estimate of 67 km s$t_{ji}=t_j-t_i$1 reproduced a directly measured multipoint instantaneous expansion speed of 68 km s$t_{ji}=t_j-t_i$2, whereas conventional time-independent methods returned smaller values of 44 km s$t_{ji}=t_j-t_i$3 and 50 km s$t_{ji}=t_j-t_i$4 (Agarwal et al., 10 Apr 2026).