BHEX Mini

BHEX Mini

Direct Imaging Black Holes from LEO

Ref Bari | 06/17 Update

\text{MEO} (20000 \text{ km})
\text{LEO} (400 \text{ km})
\textbf{BHEX}
\textbf{BHEX Mini}

Metrics and motivations for Earth–space VLBI: Time-resolving Sgr A* with the Event Horizon Telescope (Palumbo et. al., ApJ 2019)

22\mu as<\theta_{\text{BHEX-Mini}} < 1800 \mu as
5.6 G \lambda < b_{s s}<9.3 G \lambda , 0.11 G \lambda < b_{s g}<3.5 G \lambda
22\mu as<\theta_{\text{BHEX-Mini}} < 1800 \mu as
5.6 G \lambda < b_{s s}<9.3 G \lambda , 0.11 G \lambda < b_{s g}<3.5 G \lambda
R_{\max } \approx \frac{P_t G_t G_r \eta}{k T_b B}\left(N_{\bmod }\right), G_r=\eta\left(\frac{\pi D}{\lambda}\right)^2

Todd Ely

Joseph Lazio

Eric Burt

Ben Hudson

Luke Anderson

Rick Fleeter

  1. 🎯 Primary Science Objectives
  2. 🔭 SWAPC Requirements
  3. 📻 Antenna Dimensions
  4. 🧊 Cryocooler Requirements
  5. 🕰️ Frequency Reference System
  6. 🕒 BHEX Mini Timeline
  7. 💰Funding Deadlines

BHEX Mini

  1. 🎯 Primary Science Objectives
  2. 🔭 SWAPC Requirements
  3. 📻 Antenna Dimensions
  4. 🧊 Cryocooler Requirements
  5. 🕰️ Frequency Reference System
  6. 🕒 BHEX Mini Timeline
  7. 💰Funding Deadlines

BHEX Mini

🎯 Primary Science Objectives

Original Mission

Descoped Mission

Descope Mission #2

🎯 Primary Science Objectives

OG

  • 86 GHz VLBI survey of AGN Targets with d~2.5m antenna
  • Supplement (u,v) coverage of Sgr A*/M87 at 86 GHz
  • Achieve first Space-Space VLBI

Descoped Mission

Descope Mission #2

🎯 Primary Science Objectives

OG

  • 86 GHz VLBI survey of AGN Targets with d~2.5m antenna
  • Supplement (u,v) coverage of Sgr A*/M87 at 86 GHz
  • Achieve first Space-Space VLBI

DS1

  • d~2m antenna cooled down to sub-25K temperatures
  • No Frequency Phase Transfer from 86 to 230 GHz
  • ABRACON OCXO for Frequency Reference System

Descope Mission #2

🎯 Primary Science Objectives

OG

  • 86 GHz VLBI survey of AGN Targets with d~2.5m antenna
  • Supplement (u,v) coverage of Sgr A*/M87 at 86 GHz
  • Achieve first Space-Space VLBI

DS1

  • d~2m antenna cooled down to sub-25K temperatures
  • No Frequency Phase Transfer from 86 to 230 GHz
  • ABRACON OCXO for Frequency Reference System

DS2

  • 3 GHz VLBI survey of AGN targets
  • No Space-Space VLBI with BHEX
  • Ground-Space VLBI restricted to f~3 GHz ground stations

🎯 Primary Science Objectives

10 \text{ GHz}
47 \text{ GHz}
220 \text{ GHz}
1100 \text{ GHz}
\text{BHEX}
\text{BHEX Mini}
\text{EHT}
  1. 🎯 Primary Science Objectives
  2. 🔭 SWAPC Requirements
  3. 📻 Antenna Dimensions
  4. 🧊 Cryocooler Requirements
  5. 🕰️ Frequency Reference System
  6. 🕒 BHEX Mini Timeline
  7. 💰Funding Deadlines

BHEX Mini

🔭 SWAPC Requirements

Antenna

Cryocooler

Frequency Reference System

🔭 SWAPC Requirements

Antenna

  • Antenna Diameter: 
  • Primary Receiver Temperature:
  • Antenna Areal Density: 
2.2< d < 2.5m
T_r \sim 20^{\circ}K
2<\sigma<5 \text{ kg}/\text{m}^2

Cryocooler

Frequency Reference System

🔭 SWAPC Requirements

Antenna

RSP2

  • Raytheon RSP2: Sterling-PT Hybrid (450W Input)
  • Stage 1 (Sterling): 6W at 60K
  • Stage 2 (Pulse Tube): 2.1W at 20K 
  • Antenna Diameter: 
  • Primary Receiver Temperature:
  • Antenna Areal Density: 
2.2< d < 2.5m
T_r \sim 20^{\circ}K
2<\sigma<5 \text{ kg}/\text{m}^2

Frequency Reference System

🔭 SWAPC Requirements

Antenna

RSP2

  • Raytheon RSP2: Sterling-PT Hybrid (450W Input)
  • Stage 1 (Sterling): 6W at 60K
  • Stage 2 (Pulse Tube): 2.1W at 20K 

USO

  • LISA USO: 
  • RK409 Rakon USO:
  • O-CS41 ABRACON OCXO:
  • Antenna Diameter: 
  • Primary Receiver Temperature:
  • Antenna Areal Density: 
2.2< d < 2.5m
T_r \sim 20^{\circ}K
2<\sigma<5 \text{ kg}/\text{m}^2
\sigma_y = 8\cdot 10^{-15} (t=10s)
\sigma_y = 1\cdot 10^{-12} (t=10s)
\sigma_y = 2\cdot 10^{-12}(t=1s)

🔭 SWAPC Requirements

Antenna

RSP2

  • Raytheon RSP2: Sterling-PT Hybrid
  • Stage 1 (Sterling): 6W at 60K
  • Stage 2 (Pulse Tube): 2.1W at 20K 

USO

  • LISA USO: 
  • RK409 Rakon USO:
  • O-CS41 ABRACON OCXO:
  • Antenna Diameter: 
  • Primary Receiver Temperature:
  • Antenna Areal Density: 
2.2< d < 2.5m
T_r \sim 20^{\circ}K
2<\sigma<5 \text{ kg}/\text{m}^2
\sigma_y = 8\cdot 10^{-15}
\sigma_y = 1\cdot 10^{-12}
\sigma_y = 2\cdot 10^{-12}
  1. 🎯 Primary Science Objectives
  2. 🔭 SWAPC Requirements
  3. 📻 Antenna Dimensions
  4. 🧊 Cryocooler Requirements
  5. 🕰️ Frequency Reference System
  6. 🕒 BHEX Mini Timeline
  7. 💰Funding Deadlines

BHEX Mini

📻 Antenna Dimensions

3C 84

NRAO 530

NGC 1052

BL Lac

3C 273

📻 Antenna Dimensions

3C 84

NRAO 530

NGC 1052

BL Lac

3C 273

📻 Antenna Dimensions

3C 84: Nucleus of galaxy NGC 1275 (22 GHz)

NRAO 530

NGC 1052

BL Lac

3C 273

📻 Antenna Dimensions

3C 84

NRAO 530: Quasar 230 GHz, 20 μas (EHT)

NGC 1052

BL Lac

3C 273

📻 Antenna Dimensions

3C 84

NRAO 530

NGC 1052: Bright Elliptical Galaxy (65 mln lys)

BL Lac

3C 273

📻 Antenna Dimensions

3C 84

NRAO 530

NGC 1052

BL Lac

3C 279: An 'optically violent' variable quasar

Gamma Ray Image

📻 Antenna Dimensions

3C 84

NRAO 530

NGC 1052

BL Lac

3C 273

📻 Antenna Dimensions

📻 Antenna Dimensions

📻 Antenna Dimensions

📻 Antenna Dimensions

📻 Antenna Dimensions

📻 Antenna Dimensions

  1. 🎯 Primary Science Objectives
  2. 🔭 SWAPC Requirements
  3. 📻 Antenna Dimensions
  4. 🧊 Cryocooler Requirements
  5. 🕰️ Frequency Reference System
  6. 🕒 BHEX Mini Timeline
  7. 💰Funding Deadlines

BHEX Mini

🧊 Cryocooler Requirements

LT-RSP2 Cryocooler

Raytheon long life cryocoolers for future space missions (T. Conrad et. al., Cryogenics 2017)

🧊 Cryocooler Requirements

Stirling Cryocooler

Development of Advanced Two-Stage Stirling Cryocooler for Next Space Missions (Y. Sato et. al., Cryocoolers 15, 2009)

🧊 Cryocooler Requirements

Stirling Cryocooler

Development of Advanced Two-Stage Stirling Cryocooler for Next Space Missions (Y. Sato et. al., Cryocoolers 15, 2009)

🧊 Cryocooler Requirements

Stirling Cryocooler

Development of Advanced Two-Stage Stirling Cryocooler for Next Space Missions (Y. Sato et. al., Cryocoolers 15, 2009)

Phase Coherence for BHEX Mini

Phase Coherence for BHEX Mini

\Delta \phi = 2\pi \cdot f \cdot \sigma_t

Phase Error

Phase Coherence for BHEX Mini

\Delta \phi = 2\pi \cdot f \cdot \sigma_t

Observing Frequency

Phase Coherence for BHEX Mini

\Delta \phi = 2\pi \cdot f \cdot \sigma_t

Timing Jitter

Phase Coherence for BHEX Mini

\Delta \phi = 2\pi \cdot f \cdot \sigma_t

Allan Deviation

\sigma_t = \sigma_f \cdot \Delta t

Phase Coherence for BHEX Mini

\Delta \phi = 2\pi \cdot f \cdot \sigma_t

Integration Time

\sigma_t = \sigma_f \cdot \Delta t

Phase Coherence for BHEX Mini

\Delta \phi = 2\pi \cdot f \cdot \sigma_t

Integration Time

\sigma_t = \sigma_f \cdot \Delta t
\sigma_f = 5\cdot 10^{-11}

Phase Coherence for BHEX Mini

\Delta \phi = 2\pi \cdot f \cdot \sigma_t
\sigma_t = \sigma_f \cdot \Delta t
\sigma_f = 5\cdot 10^{-11}, f_{obs}=86 \text{ GHz}
\Delta t = 1s:
\sigma_t = 5\cdot 10^{-11} \cdot 1s = 5\cdot 10^{-11} s

Phase Coherence for BHEX Mini

\Delta \phi = 2\pi \cdot f \cdot \sigma_t
\sigma_t = \sigma_f \cdot \Delta t
\sigma_f = 5\cdot 10^{-11}, f_{obs}=86 \text{ GHz}
\Delta t = 1s:
\sigma_t = 5\cdot 10^{-11} \cdot 1s = 5\cdot 10^{-11} s
\Delta \phi = 2\pi \cdot (86\cdot 10^9 \text{ Hz}) \cdot 5\cdot 10^{-11} s=27.01 \text{ rad}

Phase Coherence for BHEX Mini

\Delta \phi = 2\pi \cdot f \cdot \sigma_t
\sigma_t = \sigma_f \cdot \Delta t
\sigma_f = 5\cdot 10^{-11}, f_{obs}=86 \text{ GHz}
\Delta t = 1s:
\sigma_t = 5\cdot 10^{-11} \cdot 1s = 5\cdot 10^{-11} s
\Delta \phi = 2\pi \cdot (86\cdot 10^9 \text{ Hz}) \cdot 5\cdot 10^{-11} s=27.01 \text{ rad}
\Delta t = 10s:
\sigma_t = 5\cdot 10^{-11} \cdot 10s = 50\cdot 10^{-11} s
\Delta \phi = 2\pi \cdot (86\cdot 10^9 \text{ Hz}) \cdot 50\cdot 10^{-11} s=270.01 \text{ rad}
\Delta \phi<1 \text{ rad for Phase Coherence}

Phase Coherence for BHEX Mini

Coherence Loss for BHEX Mini

L = 1-\exp\left(-2\pi^{2}f^{2}t^{2}\sigma_y^{2}\right)

Coherence Loss

Coherence Loss for BHEX Mini

Observing Frequency

L = 1-\exp\left(-2\pi^{2}f^{2}t^{2}\sigma_y^{2}\right)

Coherence Loss for BHEX Mini

Integration Time

L = 1-\exp\left(-2\pi^{2}f^{2}t^{2}\sigma_y^{2}\right)

Allan Deviation

L = 1-\exp\left(-2\pi^{2}f^{2}t^{2}\sigma_y^{2}\right)

Coherence Loss for BHEX Mini

Allan Deviation

f=86 \text{ GHz}, t=10s, \sigma_y = 5\cdot 10^{-11}

ABRACON SMD OCXO

L = 1-\exp\left(-2\pi^{2}f^{2}t^{2}\sigma_y^{2}\right)
L = 1-\exp\left(-2\pi^{2}f^{2}t^{2}\sigma_y^{2}\right)

Coherence Loss for BHEX Mini

Allan Deviation

f=86 \text{ GHz}, t=10s, \sigma_y = 5\cdot 10^{-11}
L = 1-\exp\left[-2\pi^{2}(86\cdot 10^9)^{2}(10)^{2}(5\cdot 10^{-11})^{2}\right]

ABRACON SMD OCXO

L = 1-\exp\left(-2\pi^{2}f^{2}t^{2}\sigma_y^{2}\right)

Coherence Loss for BHEX Mini

Allan Deviation

f=86 \text{ GHz}, t=10s, \sigma_y = 5\cdot 10^{-11}
L = 1-\exp\left[-2\pi^{2}(86\cdot 10^9)^{2}(10)^{2}(5\cdot 10^{-11})^{2}\right]

ABRACON SMD OCXO

L = 1-\exp\left(-2\pi^{2}f^{2}t^{2}\sigma_y^{2}\right)
L\sim 1\%<10\% \text{ required for Phase Coherence}

Antenna Diameter for BHEX Mini

🕒 Prospective Timeline

June

July

August

September

🕒 Prospective Timeline

June

July

August

September

  • NASA NIAC 2025 Phase I Step I
  • SpaceCom Conference 2026
  • Brown Nelson + Hazeltine Grants
  • Antenna Focus
    • Nacer Chahat
    • Emmanuel Decrossas

🕒 Prospective Timeline

June

July

August

September

  • Cryocooler Focus
    • Lucas Anderson
    • Katelyn Boushon
  • NSF Foundational Research in Robotics Grant (FRR)
  • Fall Walls Foundation Selections
  • NASA NIAC 2025 Phase I Step I
  • SpaceCom Conference 2026
  • Brown Nelson + Hazeltine Grants
  • Antenna Focus
    • Nacer Chahat
    • Emmanuel Decrossas

🕒 Prospective Timeline

June

July

Aug

September

  • Cryocooler Focus
    • Lucas Anderson
    • Katelyn Boushon
  • NSF Foundational Research in Robotics Grant (FRR)
  • Fall Walls Foundation Selections
  • Brown University Co-Lab 
  • NASA NIAC Phase I Round I Step B Selections Announced
  • Solar Panel Focus
    • DCubed Inc.
    • DHV Tech
  • NASA NIAC 2025 Phase I Step I
  • SpaceCom Conference 2026
  • Brown Nelson + Hazeltine Grants
  • Antenna Focus
    • Nacer Chahat
    • Emmanuel Decrossas

🕒 Prospective Timeline

June

July

Aug

Sep

  • Cryocooler Focus
    • Lucas Anderson
    • Katelyn Boushon
  • NSF Foundational Research in Robotics Grant (FRR)
  • Fall Walls Foundation Selections
  • Brown University Co-Lab 
  • NASA NIAC Phase I Round I Step B Selections Announced
  • Solar Panel Focus
    • DCubed Inc.
    • DHV Tech
  • NSF Advanced Technologies and Instrumentation for the Astronomical Sciences (ATI) 
  • Data Downlink Focus
    • MIT Lincoln Labs
    • ALICE/CLICK Teams
  • NASA NIAC 2025 Phase I Step I
  • SpaceCom Conference 2026
  • Brown Nelson + Hazeltine Grants
  • Antenna Focus
    • Nacer Chahat
    • Emmanuel Decrossas

🕒 Prospective Timeline

Antenna

🕒 Prospective Timeline

Cryocooler

🕒 Prospective Timeline

Solar Panels

🕒 Prospective Timeline

Orbital Parameters

🕒 Prospective Timeline

Data Downlink

🕒 Prospective Timeline

Systems Integration

  1. 🕒 Prospective Timeline
  2. 💰Funding Deadlines
  3. 🎯 Primary Science Objectives
  4. 🔭 Antenna Requirements
  5. 🧠 Ideas!

BHEX Mini

💰Funding Oppurtunities

June

$3,000

💰Funding Oppurtunities

July

NASA NIAC Phase I

$175,000

$3,000

$175,000

💰Funding Oppurtunities

Sep

NASA NIAC Phase I

$175,000

$3,000

$250,000

💰Funding Oppurtunities

NASA NIAC Phase I

Oct

$175,000

$3,000

$250,000

💰Funding Oppurtunities

  1. 🕒 Prospective Timeline
  2. 💰Funding Deadlines
  3. 🎯 Primary Science Objectives
  4. 🔭 Antenna Requirements
  5. 🧠 Ideas!

BHEX Mini

\textbf{Primary Science Objective: }\text{Resolve extended black hole structure}
\text{Image black hole accretion disk at low frequency}
\text{Increase number of short Space-Ground baselines}
\text{Achieve first Space-Space VLBI}

🎯 Mission Statement

10 \text{ GHz}
47 \text{ GHz}
220 \text{ GHz}
1100 \text{ GHz}

🎯 Mission Statement

10 \text{ GHz}
47 \text{ GHz}
220 \text{ GHz}
1100 \text{ GHz}
\text{BHEX}
\text{BHEX Mini}
\text{EHT}

🎯 Mission Statement

\textbf{BHEX Mini}
\text{PR: 86 GHz}
\textbf{BHEX}
\text{PR: 230-345 GHz}\\ \text{SR: 86-120 GHz}
  1. 🕒 Prospective Timeline
  2. 💰Funding Deadlines
  3. 🎯 Primary Science Objectives
  4. 🔭 Antenna Requirements
  5. 🧠 Ideas!

BHEX Mini

D
D\sim2 m
T\sim20^{\circ}K

Secondary 

Science Targets

Gaussian Source Approximation

Visibility Amplitudes

Thermal Noise Constraints

\sigma_{GS}=\frac{1}{\eta_{\mathrm{Q}}} \sqrt{\frac{\mathrm{SEFD}_{\mathrm{G}} \mathrm{SEFD}_{\mathrm{S}}}{2 \Delta \nu \Delta t}}

SEFD Constraint

\sigma_{GS}=\frac{1}{\eta_{\mathrm{Q}}} \sqrt{\frac{\mathrm{SEFD}_{\mathrm{G}} \mathrm{SEFD}_{\mathrm{S}}}{2 \Delta \nu \Delta t}}
\mathrm{SEFD}_S=\frac{2 k_{\mathrm{B}} T_{\mathrm{sys}}^*}{\eta_{\mathrm{A}} A}

Constraints

T,A
\sigma_{GS}=\frac{1}{\eta_{\mathrm{Q}}} \sqrt{\frac{\mathrm{SEFD}_{\mathrm{G}} \mathrm{SEFD}_{\mathrm{S}}}{2 \Delta \nu \Delta t}}
\mathrm{SEFD}_S=\frac{2 k_{\mathrm{B}} T_{\mathrm{sys}}^*}{\eta_{\mathrm{A}} A}

 

Parameter Space

\sigma, T,A
\sigma_{GS}=\frac{1}{\eta_{\mathrm{Q}}} \sqrt{\frac{\mathrm{SEFD}_{\mathrm{G}} \mathrm{SEFD}_{\mathrm{S}}}{2 \Delta \nu \Delta t}}
D
\mathrm{SEFD}_S=\frac{2 k_{\mathrm{B}} T_{\mathrm{sys}}^*}{\eta_{\mathrm{A}} A}

Antenna Diameter + Temperature!

Visibility Amplitudes

Visibility Amplitudes

Visibility Amplitudes

Visibility Amplitudes

Visibility Amplitudes

Delta

Visibility Amplitudes

Delta

Visibility Amplitudes

Delta

Visibility Amplitudes

Delta

Visibility Amplitudes

Delta

V(b) = S\cdot \exp(-\pi^2 b^2\theta^2/4\ln 2)
V(b) = S\cdot \exp(-\pi^2 b^2\theta^2/4\ln 2)
\sigma_{SS}<\frac{|V_{SS}|}{SNR}, \sigma_{SG}<\frac{|V_{SG}|}{SNR}
\sigma_{SS}<\frac{|V_{SS}|}{SNR}, \sigma_{SG}<\frac{|V_{SG}|}{SNR}

BHEX Mini

  1. 🕒 Prospective Timeline
  2. 💰Funding Deadlines
  3. 🎯 Primary Science Objectives
  4. 🔭 Antenna Requirements
  5. 🧠 Introduction to (u,v) Plane

🧠 The (u,v) Plane

🧠 The (u,v) Plane

🧠 The (u,v) Plane

🧠 The (u,v) Plane

🧠 The (u,v) Plane

🧠 The (u,v) Plane

🧠 The (u,v) Plane

🧠 The (u,v) Plane

Goal:

I(l,m)

🧠 The (u,v) Plane

I(l,m)

Intensity of a certain part of the sky/

sky brightness pattern

Goal: Measure

🧠 The (u,v) Plane

Goal: Measure

I(l,m)

Coordinates in the sky

🧠 The (u,v) Plane

Now we add the effects of the radio interferometer ...

I(l,m)

Coordinates in the sky

🧠 The (u,v) Plane

Now we add the effects of the radio interferometer ...

I(l,m)

Multiplicative envelope:

comes from size of antennas

A(l,m)

🧠 The (u,v) Plane

Now we add the effects of the radio interferometer ...

I(l,m)

Simulates interference pattern

A(l,m)
e^{-2\pi i (ul+vm)}

🧠 The (u,v) Plane

Now we add the effects of the radio interferometer ...

I(l,m)

(u,v): Baseline vector

u: East-west baseline distance

v: North-south baseline distance

A(l,m)
e^{-2\pi i (ul+vm)}
\vec{b}=(u,v)

🧠 The (u,v) Plane

Take all the signals from the sky and add them up ...

I(l,m)
A(l,m)
e^{-2\pi i (ul+vm)}
\int
dl dm
V(u,v)=

🧠 The (u,v) Plane

Take all the signals from the sky and add them up ...

I(l,m)
A(l,m)
e^{-2\pi i (ul+vm)}
\int
dl dm
V(u,v)=

Visibility Function

🧠 The (u,v) Plane

Take all the signals from the sky and add them up ...

I(l,m)
A(l,m)
e^{-2\pi i (ul+vm)}
\int
dl dm
V(u,v)=

Multiplicative Envelope

🧠 The (u,v) Plane

Take all the signals from the sky and add them up ...

I(l,m)
A(l,m)
e^{-2\pi i (ul+vm)}
\int
dl dm
V(u,v)=

Sky Intensity/Brightness

🧠 The (u,v) Plane

Take all the signals from the sky and add them up ...

I(l,m)
A(l,m)
e^{-2\pi i (ul+vm)}
\int
dl dm
V(u,v)=

Interferometric Pattern

\vec{b}=(u,v)

🧠 The (u,v) Plane

It's a Fourier Transformation!

I(l,m)
A(l,m)
e^{-2\pi i (ul+vm)}
\int
dl dm
V(u,v)=

One pair of antennas measures one single point on the (u,v) plane: one fourier mode!

🧠 The (u,v) Plane

It's a Fourier Transformation!

I(l,m)
A(l,m)
e^{-2\pi i (ul+vm)}
\int
dl dm
V(u,v)=

What we want

🧠 The (u,v) Plane

It's a Fourier Transformation!

I(l,m)
A(l,m)
e^{-2\pi i (ul+vm)}
\int
dl dm
V(u,v)=

What we get :(

🧠 The (u,v) Plane

It's a Fourier Transformation!

Fill up the (u,v) plane

and then fourier transform back to the real image!

🧠 The (u,v) Plane

It's a Fourier Transformation!

Fill up the (u,v) plane

and then fourier transform back to the real image!

🧠 The (u,v) Plane

But how do you fill up the (u,v) plane?

🧠 The (u,v) Plane

But how do you fill up the (u,v) plane?

  1. Use an array of antennas!
N \text{ antennas} = \frac{N(N-1)}{2} \text{ baselines}

🧠 The (u,v) Plane

  1. Use an array of antennas!
N \text{ antennas} = \frac{N(N-1)}{2} \text{ baselines}

🧠 The (u,v) Plane

  1. Use an array of antennas!
4 \text{ antennas} = \frac{4(3)}{2}=6 \text{ baselines}

🧠 The (u,v) Plane

But how do you fill up the (u,v) plane?

  1. Use an array of antennas!
N \text{ antennas} = \frac{N(N-1)}{2} \text{ baselines}

2. Earth Rotation Aperture Synthesis

\vec{u} = \frac{\vec{b}\cdot \hat{s}}{\lambda}

🧠 The (u,v) Plane

2. Earth Rotation Aperture Synthesis

\vec{u} = \frac{\vec{b}\cdot \hat{s}}{\lambda}
\vec{b}\cdot \hat{s}