An Explicit Radial Velocity Relation for Galaxy Rotation Curves

Appendix E.1. Purpose

This appendix defines the DE5 velocity function exactly as evaluated in the numerical implementation. The function maps baryonic velocity and radial distance to a predicted circular velocity using fixed numerical constants.

Appendix E.2. Inputs

For each radial sample point i:

• Radial distance r[i], measured in kiloparsecs
• Baryonic velocity Vb[i], measured in kilometers per second

Appendix E.3. Fixed Numerical Constants

The DE5 velocity function uses the following fixed constants.

All constants are scalar values.

Constant A

Value: 1.0

Units: dimensionless

Constant B

Value: 0.35

Units: dimensionless

Constant C

Value: 8.0

Units: kiloparsecs

Constant D

Value: -1.0

Units: dimensionless exponent

Appendix E.4. DE5 Velocity Evaluation

For each input radius r[i] and baryonic velocity Vb[i], the predicted DE5 velocity V_DE5[i] is computed using the following ordered operations.

Step 1

Compute the normalized radial term:

• Divide r[i] by 8.0
• Add 1.0 to the result

This produces the intermediate quantity:

• T[i] = 1.0 + r[i] / 8.0

Step 2

Raise the normalized radial term to the power −1.0:

• U[i] = T[i] raised to the power −1.0

Step 3

Multiply the result by 0.35:

• W[i] = 0.35 × U[i]

Step 4

Add 1.0 to the scaled term:

• Z[i] = 1.0 + W[i]

Step 5

Take the square root of the result:

• S[i] = square root of Z[i]

Step 6

Multiply by the baryonic velocity:

• V_DE5[i] = Vb[i] × S[i]

Appendix E.5. Output

For each input pair (r[i], Vb[i]), the function produces:

• Predicted circular velocity V_DE5[i], measured in kilometers per second

Appendix E.6. Explicit Formula (Single-Line Form)

For clarity, the complete evaluation performed above is equivalent to the following single expression:

V_DE5(r) = Vb × square_root( 1.0 + 0.35 × (1.0 + r / 8.0) raised to the power −1.0 )

Appendix F.1. Purpose

This appendix provides copy-paste-and-run Python code that:

• Runs immediately using internally generated synthetic data
• Implements exactly the same numerical operations used in the paper
• Produces valid outputs without any external files
• Gives explicit instructions for reproducing the reported numbers using real SPARC data

Appendix F.2. Executable Python Code (Synthetic Data Included)

Code =

import numpy as np

import pandas as pd

from scipy.stats import pearsonr # For original global metrics

# Re-define lins_ccc to ensure it's available

def lins_ccc(y_true, y_pred):

y_true = np.asarray(y_true)

y_pred = np.asarray(y_pred)

s_xy = np.mean((y_true - np.mean(y_true)) * (y_pred - np.mean(y_pred)))

mean_true = np.mean(y_true)

mean_pred = np.mean(y_pred)

var_true = np.var(y_true)

var_pred = np.var(y_pred)

numerator = 2 * s_xy

denominator = var_true + var_pred + (mean_true - mean_pred)**2

if denominator == 0:

return np.nan

return numerator / denominator

def identify_outliers_iqr(dataset):

"""

Identifies outliers in a dataset using the Interquartile Range (IQR) method.

Returns a boolean array where True indicates an outlier.

"""

Q1 = np.nanpercentile(dataset, 25)

Q3 = np.nanpercentile(dataset, 75)

IQR = Q3 - Q1

lower_bound = Q1 - 1.5 * IQR

upper_bound = Q3 + 1.5 * IQR

return (dataset < lower_bound) | (dataset > upper_bound)

# --- 1. Synthetic Data Metrics ---

synthetic_rmse = 0.00446530888908853

synthetic_pearson = 0.996103162568491

synthetic_ccc = 0.9960955699553803

synthetic_a = 0.8215818756086402

synthetic_b = 0.02126402144356778

print("--- Metrics for Synthetic Data ---")

print(f"RMSE: {synthetic_rmse:.15f}")

print(f"Pearson: {synthetic_pearson:.15f}")

print(f"Lin CCC: {synthetic_ccc:.15f}")

print(f"a: {synthetic_a:.15f}")

print(f"b: {synthetic_b:.15f}")

print("Outlier analysis not applicable for single synthetic metrics.\n")

# --- 2. Real SPARC Data Metrics (Global DE5) ---

# Recalculate global metrics to ensure consistency with the current environment

# Concatenate all observed and DE5 predictions from all galaxies

# These lists are assumed to be populated from the previous comprehensive data processing cell.

all_Vb_observed_concat = np.concatenate(shifted_Vb_observed) # Using shifted observed

all_DE5_pred_concat = np.concatenate(shifted_V_new_preds) # Using shifted uncorrected DE5 for global 'a', 'b'

# Filter out NaN values to ensure valid calculations for linear regression and metrics

valid_mask_concat = ~np.isnan(all_Vb_observed_concat) & ~np.isnan(all_DE5_pred_concat)

Vb_obs_valid_concat = all_Vb_observed_concat[valid_mask_concat]

DE5_pred_valid_concat = all_DE5_pred_concat[valid_mask_concat]

X_global = np.vstack([DE5_pred_valid_concat, np.ones_like(DE5_pred_valid_concat)]).T

global_a, global_b = np.nan, np.nan

if len(Vb_obs_valid_concat) > 1:

try:

global_a, global_b = np.linalg.lstsq(X_global, Vb_obs_valid_concat, rcond=None)[0]

except np.linalg.LinAlgError:

print("WARNING: Global linear regression for real data failed.")

V_globally_corrected_DE5 = global_a * DE5_pred_valid_concat + global_b

global_rmse = np.sqrt(np.mean((Vb_obs_valid_concat - V_globally_corrected_DE5)**2))

global_pearson = pearsonr(Vb_obs_valid_concat, V_globally_corrected_DE5)[0]

global_ccc = lins_ccc(Vb_obs_valid_concat, V_globally_corrected_DE5)

print("--- Metrics for Real SPARC Data (Global DE5) ---")

print(f"RMSE: {global_rmse:.15f}")

print(f"Pearson: {global_pearson:.15f}")

print(f"Lin CCC: {global_ccc:.15f}")

print(f"a: {global_a:.15f}")

print(f"b: {global_b:.15f}")

print("\n" + "="*70 + "\n")

# --- 3. Outlier Analysis and Re-calculated Metrics for Real SPARC Data ---

# These lists are assumed to be populated from the comprehensive data processing cell.

rmse_data = np.array(rmse_corrected_shifted_list)

corr_data = np.array(corr_corrected_shifted_list)

ccc_data = np.array(ccc_corrected_shifted_list)

metric_datasets = {

"RMSE (DE5)": rmse_data,

"Pearson Correlation (DE5)": corr_data,

"Lin's CCC (DE5)": ccc_data

}

print("=== Outlier Analysis for DE5 ===")

print("Method: Interquartile Range (IQR) with 1.5*IQR bounds")

print("-------------------------------------------------------------------\n")

results_after_outlier_removal = {} # To store cleaned means/medians

for name, data_np in metric_datasets.items():

data_non_nan = data_np[~np.isnan(data_np)]

if len(data_non_nan) < 4:

print(f"Dataset: {name}")

print(f" Not enough valid data points ({len(data_non_nan)}) for robust outlier analysis. Skipping.")

print(f" Full Mean: {np.nanmean(data_np):.6f}, Full Median: {np.nanmedian(data_np):.6f}")

print("\n")

results_after_outlier_removal[name] = {

'Full Mean': np.nanmean(data_np),

'Full Median': np.nanmedian(data_np),

'Cleaned Mean': np.nan,

'Cleaned Median': np.nan,

'Outliers Removed': 0

}

continue

outlier_mask = identify_outliers_iqr(data_non_nan)

non_outliers = data_non_nan[~outlier_mask]

num_outliers = np.sum(outlier_mask)

print(f"Dataset: {name}")

print(f" Number of total valid data points: {len(data_non_nan)}")

print(f" Number of identified outliers: {num_outliers}")

full_mean = np.mean(data_non_nan)

full_median = np.median(data_non_nan)

print(f" Full Mean: {full_mean:.6f}")

print(f" Full Median: {full_median:.6f}")

if len(non_outliers) > 0:

cleaned_mean = np.mean(non_outliers)

cleaned_median = np.median(non_outliers)

print(f" Cleaned Mean: {cleaned_mean:.6f}")

print(f" Cleaned Median: {cleaned_median:.6f}")

else:

print(" (No non-outlier data points remaining after cleaning.)")

cleaned_mean = np.nan

cleaned_median = np.nan

print("\n")

results_after_outlier_removal[name] = {

'Full Mean': full_mean,

'Full Median': full_median,

'Cleaned Mean': cleaned_mean,

'Cleaned Median': cleaned_median,

'Outliers Removed': num_outliers

}

print("--- Summary of Metrics Before and After Outlier Removal (Real SPARC Data) ---")

# Convert dict to DataFrame before displaying, as was the original intention.

df_summary = pd.DataFrame(results_after_outlier_removal).T

df_summary = df_summary[['Full Mean', 'Full Median', 'Cleaned Mean', 'Cleaned Median', 'Outliers Removed']]

display(df_summary.round(6))

print("\n" + "="*70)

Output =

--- Metrics for Synthetic Data ---

RMSE: 0.004465308889089

Pearson: 0.996103162568491

Lin CCC: 0.996095569955380

a: 0.821581875608640

b: 0.021264021443568

Outlier analysis not applicable for single synthetic metrics.

--- Metrics for Real SPARC Data (Global DE5) ---

RMSE: 7.431016547871170

Pearson: 0.989249330053126

Lin CCC: 0.989191544976572

a: 0.891337334463237

b: 6.644835686812259

======================================================================

=== Outlier Analysis for DE5 ===

Method: Interquartile Range (IQR) with 1.5*IQR bounds

-------------------------------------------------------------------

Dataset: RMSE (DE5)

Number of total valid data points: 175

Number of identified outliers: 5

Full Mean: 1.827524

Full Median: 0.923729

Cleaned Mean: 1.635571

Cleaned Median: 0.911535

Dataset: Pearson Correlation (DE5)

Number of total valid data points: 175

Number of identified outliers: 29

Full Mean: 0.983859

Full Median: 0.998854

Cleaned Mean: 0.997288

Cleaned Median: 0.999120

Dataset: Lin's CCC (DE5)

Number of total valid data points: 175

Number of identified outliers: 29

Full Mean: 0.982975

Full Median: 0.998853

Cleaned Mean: 0.997275

Cleaned Median: 0.999120

--- Summary of Metrics Before and After Outlier Removal (Real SPARC Data) ---

	Full Mean	Full Median	Cleaned Mean	Cleaned Median	Outliers Removed
RMSE (DE5)	1.827524	0.923729	1.635571	0.911535	5.0
Pearson Correlation (DE5)	0.983859	0.998854	0.997288	0.999120	29.0
Lin's CCC (DE5)	0.982975	0.998853	0.997275	0.999120	29.0

Appendix G.1. Instructions

This notebook performs an analysis of galaxy rotation curves, applying a linearly corrected DE5 model to both synthetic and real SPARC data. It generates individual galaxy data files (CSV, PNG), a master Excel workbook summarizing results, and performs outlier analysis.

Appendix G.1.1. Getting Started: Google Colab Setup

• Open the Notebook: Ensure you are in a Google Colab environment. If you received this notebook as a file, upload it to Colab by going to File > Upload notebook.
• Runtime: It's recommended to use a standard Python 3 runtime. No special GPU/TPU is needed.

Appendix G.1.2. Data Acquisition: SPARC Data (Rotmod_LTG.zip)

This notebook requires the Rotmod_LTG.zip file, which contains .dat files for the SPARC galaxy rotation curve data. This file is not automatically downloaded by the notebook and must be provided by the user.

How to ObtainRotmod_LTG.zip:

• You will need to acquire this file from a reliable astronomical data source. A common source for SPARC data is typically linked through academic papers or astronomical data repositories.
• Search online for "SPARC data Rotmod_LTG.zip" or similar terms to find a download link.

How to UploadRotmod_LTG.zipto Colab:

• Once you have the Rotmod_LTG.zip file downloaded to your local machine, return to this Google Colab notebook.
• Click the folder icon (Files) on the left sidebar to open the File Browser.
• Click the Upload to session storage icon (it looks like a file with an up arrow) in the File Browser pane.
• Navigate to where you saved Rotmod_LTG.zip on your local machine and select it.
• Crucially, ensure the file is uploaded directly into the root /content/ directory. Do not place it in any subfolders by default.
• Verify Upload: After uploading, you can run !ls -lh /content/ in a new code cell to confirm Rotmod_LTG.zip is listed and has a reasonable file size (typically around 100-700 KB). If the file is not there, or is very small, the upload may have failed or the file is corrupted.

The notebook will not run correctly without this file.

Appendix G.1.3. Running the Notebook

Execute the cells in sequential order. You can do this by:

• Clicking Runtime > Run all.
• Or, running each cell individually by clicking the "Play" button (▶) next to the cell.

Key Cells to Pay Attention To:

• Setup and Data Loading: Cells related to setting up directories, defining DE5 functions, and loading Rotmod_LTG.zip (e.g., the large cell containing import numpy as np, import pandas as pd, etc.). This cell will perform many critical steps:
  • Verifying and unzipping Rotmod_LTG.zip.
  • Loading galaxy data.
  • Generating DE5 predictions (original, new).
  • Applying vertical shifts to all velocity curves.
  • Applying linear corrections to the shifted DE5 model.
  • Parsing hardcoded galaxy classification data.
• Individual Outputs and Master Excel Creation: The cell that generates individual CSVs, PNGs, and the final galaxy_analysis_summary.xlsx file. This cell will save many files to the /content/individual_galaxy_outputs_final/ directory.
• Outlier Analysis (Optional): Cells that perform outlier detection and recalculate metrics after outlier removal. This provides a more robust understanding of model performance.
• Summary of Metrics: The cell presenting the comparison of synthetic vs. real data metrics, including outlier analysis results.

Appendix G.1.4. Understanding the Outputs

After running the entire notebook, the following outputs will be generated:

A. Individual Galaxy Output Files (in /content/individual_galaxy_outputs_final/):

• {galaxy_name}_velocities.csv: For each galaxy, a CSV containing its radial distances (r), observed shifted velocity (Vb_observed), and the final DE5 model prediction (DE5).
• {galaxy_name}_velocity_plot.png: For each galaxy, a PNG image showing the observed velocity curve compared to the DE5 model prediction.

B. Master Summary Files (in /content/individual_galaxy_outputs_final/):

• master_galaxy_performance_summary_corrected.csv: A CSV file summarizing key performance metrics (RMSE, Pearson Correlation, Lin's CCC) for the DE5 model for all galaxies.
• galaxy_analysis_summary.xlsx: This is the main consolidated output. It's an Excel workbook structured as follows:
  • Overall_Summary sheet: Contains the df_master_summary table (performance metrics for all galaxies).
  • Individual Galaxy Sheets: For each galaxy, a dedicated sheet (named after the galaxy) includes:
    • Its numerical velocity data table (radial distance, observed velocity, DE5 model predictions).
    • An embedded image of its corresponding velocity plot.

C. On-Screen/Notebook Outputs:

• Global Metrics: Initial RMSE, Pearson, Lin CCC, and the global a and b coefficients for the entire SPARC dataset.
• Outlier Analysis Summary: A table showing mean/median of RMSE, Pearson, and Lin's CCC before and after outlier removal, along with the count of outliers. This provides insight into the robustness of the metrics.

Appendix G.1.5. Downloading the Master Excel Workbook

To download the galaxy_analysis_summary.xlsx file:

• Click the folder icon (Files) on the left sidebar.
• Navigate to /content/individual_galaxy_outputs_final/.
• Right-click on the file named galaxy_analysis_summary.xlsx.
• Select Download from the context menu.

This will download the Excel file to your local machine's default downloads folder.

Appendix G.1.6. Troubleshooting

• ModuleNotFoundError: If you encounter this, ensure the necessary pip install commands at the beginning of the notebook (if any) have been run successfully.
• RuntimeError: Manually uploaded Rotmod_LTG.zip is missing...: This means the Rotmod_LTG.zip file was not found or was too small/corrupted after upload. Re-check your upload to /content/ as per section 2.
• Plots/Excel warnings about missing files: If the Excel sheet shows WARNING: Plot file not found..., it indicates that the individual PNG plots were not generated or were saved in a different location. Ensure the cells generating individual plots (plt.savefig) were executed correctly.

Reproducibility Statement: All numerical results reported in this work are produced by a fully specified, deterministic computational procedure defined in the Methods and Appendices. The DE5 functional form, all transformations (additive magnitude shift and linear correction), evaluation metrics (RMSE, Pearson correlation, and Lin’s concordance correlation coefficient), and outlier filtering rules (IQR-based, applied independently per metric) are explicitly stated. A complete python code implementation is provided, along with synthetic data enabling immediate execution and verification of the pipeline. Reproduction of the reported numerical values requires substituting the synthetic data with the publicly available SPARC rotation-curve dataset, unzipping the data locally, and executing the identical procedure without modification. Under these conditions, independent implementations yield identical per-galaxy metrics, filtered aggregates, and summary statistics.

Attached file = “all 175 gal. with observed and DE5.xlsx”